Recent zbMATH articles in MSC 16https://zbmath.org/atom/cc/162021-01-08T12:24:00+00:00WerkzeugUniform and hollow modules over rings of Morita contexts.https://zbmath.org/1449.160172021-01-08T12:24:00+00:00"Zhang, Wenhui"https://zbmath.org/authors/?q=ai:zhang.wenhui"Zhang, Li"https://zbmath.org/authors/?q=ai:zhang.li.5|zhang.li.4|zhang.li.3|zhang.li.2|zhang.li.6|zhang.li.8|zhang.li.10|zhang.li.9|zhang.li.1|zhang.li.11|zhang.li.12|zhang.li|zhang.li.7Summary: By using the decomposition of the (right)modules over rings of Morita contexts, we discuss the structures of essential submodules and small submodules of modules over these rings. For the Morita contexts \(T = \begin{pmatrix}R&M\\ N&S\end{pmatrix}_{(\theta, \psi)}\), every right \(T\)-module can be decomposed into a quadriad \( (P, Q)_{(f, g)}\), we give characterizations of the structure of uniform modules and hollow modules over these rings and the necessary conditions for \( (P, Q)_{(f, g)}\) to be uniform (hollow). Let \(L = \{p\in P|g (p \bigotimes m) = 0, \text{for all }m \in M\}\), \(K = \{q \in Q|g (q \bigotimes n) = 0, \text{for all }n \in N\}\), we prove that: (1) If \(P = 0\), and \(K = Q\) are uniform modules (or \(Q = 0\), and \(P = L\) are uniform modules), then \( (P, Q)_{(f, g)}\) is an uniform module; (2) If \(P\) and \(Q\) are hollow modules, \(f (Q \bigotimes N) = P\), \(g (P \bigotimes M) \ne Q\) (or \(f (Q \bigotimes N) \ne P\), \(g (P \bigotimes M) = Q\)), then \( (P, Q)_{(f, g)}\) is a hollow module.Non-global nonlinear Lie triple derivable maps on factor von Nuemann algebras.https://zbmath.org/1449.160852021-01-08T12:24:00+00:00"Su, Yutian"https://zbmath.org/authors/?q=ai:su.yutian"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(M\) be a factor von Neumann algebra with dimension greater than 1 on a Hilbert space \(H\). With the help of algebraic decomposition method, we prove that if a nonlinear map \(\delta:M \to M\) satisfied \(\delta ([[A, B], C]) = [[\delta (A), B], C] + [[A, \delta (B)], C] + [[A, B], \delta (C)]\) for any \(A\), \(B\), \(C \in M\) with \(ABC = 0\), then there existed an additive derivation \(d:M \to M\), such that \(\delta (A) = d (A) + \tau (A)I\) for any \(A \in M\), where \(\tau :M \to \mathbb{C}I\) is a nonlinear map such that \(\tau ([[A, B], C]) = 0\) with \(ABC = 0\) for any \(A\), \(B\), \(C \in M\).On graph associated to co-ideals of commutative semirings.https://zbmath.org/1449.160992021-01-08T12:24:00+00:00"Talebi, Yahya"https://zbmath.org/authors/?q=ai:talebi.yahya"Darzi, Atefeh"https://zbmath.org/authors/?q=ai:darzi.atefehSummary: Let \(R\) be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph \(\Omega(R)\) whose vertices are all elements of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if the product of the co-ideals generated by \(x\) and \(y\) is \(R\). Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph \(\Gamma(R)\) and \(\Omega(R)\).Construction of fusion rings from a generalized Cartan matrix of indefinite type.https://zbmath.org/1449.160522021-01-08T12:24:00+00:00"Xue, Lei"https://zbmath.org/authors/?q=ai:xue.lei"Yi, Xiaomeng"https://zbmath.org/authors/?q=ai:yi.xiaomeng"Wang, Zhihua"https://zbmath.org/authors/?q=ai:wang.zhihuaSummary: Based on some properties of fusion ring, two types of fusion rings are constructed from a given generalized Cartan matrix of indefinite type. It turns out that the two fusion rings are both near-group rings.Galois linear maps and their construction.https://zbmath.org/1449.160602021-01-08T12:24:00+00:00"Gu, Yue"https://zbmath.org/authors/?q=ai:gu.yue"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.29"Wang, Shuanhong"https://zbmath.org/authors/?q=ai:wang.shuanhongSummary: The condition of an algebra to be a Hopf algebra or a Hopf (co)quasigroup can be determined by the properties of Galois linear maps. For a bialgebra \(H\), if it is unital and associative as an algebra and counital coassociative as a coalgebra, then the Galois linear maps \({T_1}\) and \({T_2}\) can be defined. For such a bialgebra \(H\), it is a Hopf algebra if and only if \({T_1}\) is bijective. Moreover, \(T_1^{-1}\) is a right \(H\)-module map and a left \(H\)-comodule map (similar to \({T_2}\)). On the other hand, for a unital algebra (no need to be associative), and a counital coassociative coalgebra \(A\), if the coproduct and counit are both algebra morphisms, then sufficient and necessary condition of \(A\) to be a Hopf quasigroup is that \({T_1}\) is bijective, and \(T_1^{-1}\) is left compatible with \(\Delta_{T_1^{-1}}^r\) and right compatible with \(\Delta_{T_1^{-1}}^r\) at the same time (the properties are similar to \({T_2}\)). Furthermore, as a corollary, the quasigroup case is also considered.On pair of generalized derivations in rings.https://zbmath.org/1449.160762021-01-08T12:24:00+00:00"Ali, Asma"https://zbmath.org/authors/?q=ai:ali.asma"Rahaman, Md. Hamidur"https://zbmath.org/authors/?q=ai:rahaman.md-hamidurSummary: Let \(R\) be an associative ring with extended centroid \(C\), let \(G\) and \(F\) be generalized derivations of \(R\) associated with nonzero derivations \(\delta\) and \(d\), respectively, and let \(m, k, n \geq1\) be fixed integers. In the present paper, we study the situations: (i) \(F(x)\circ_mG(y)=(x \circ_n y)^k\), (ii) \([F(x),y]_m+[x,d(y)]_n=0\) for all \(y\), \(x\) in some appropriate subset of \(R\).Almost strongly regular rings and their extensions.https://zbmath.org/1449.160252021-01-08T12:24:00+00:00"Wang, Yao"https://zbmath.org/authors/?q=ai:wang.yao"Yang, Zhen"https://zbmath.org/authors/?q=ai:yang.zhen"Ren, Yanli"https://zbmath.org/authors/?q=ai:ren.yanliSummary: We introduce in this paper the concept of almost strongly regular rings on the basis of strong regular rings. These rings lie between local rings and VNL rings. We present some examples of almost strongly regular rings and investigate their extensions.Isomorphism of commutative group algebras of finite abelian \(p\)-groups.https://zbmath.org/1449.200032021-01-08T12:24:00+00:00"Mollov, Todor Zh."https://zbmath.org/authors/?q=ai:mollov.todor-zh"Nachev, Nako A."https://zbmath.org/authors/?q=ai:nachev.nako-aSummary: Let \(R\) be a direct product of commutative indecomposable rings with identities and let \(G\) be a finite abelian \(p\)-group. In the present paper we give a complete system of invariants of the group algebra \(RG\) of \(G\) over \(R\) when \(p\) is an invertible element in \(R\). These investigations extend some classical results of \textit{S. D. Berman} [Dokl. Akad. Nauk SSSR, n. Ser. 91, 7--9 (1953; Zbl 0050.25504) and Mat. Sb., Nov. Ser. 44(86), 409--456 (1958; Zbl 0080.02102)], \textit{S. K. Sehgal} [J. Number Theory 2, 500--508 (1970; Zbl 0209.05804)] and \textit{G. Karpilovsky} [Period. Math. Hung. 15, 259--265 (1984; Zbl 0526.20004)] as well as a result of the first author [PLISKA, Stud. Math. Bulg. 8, 54--64 (1986; Zbl 0655.16004)].Gorenstein flat modules and Frobenius extensions.https://zbmath.org/1449.160302021-01-08T12:24:00+00:00"Ren, Wei"https://zbmath.org/authors/?q=ai:ren.wei.1|ren.wei.3|ren.wei.2|ren.wei.4|ren.wei.5Summary: Let \(R \subset A\) be a Frobenius extension of rings, where \(A\) is right coherent. Let \(M\) be any left \(A\)-module. We first show that \(_AM\) is Gorenstein flat if and only if the underlying \(R\)-module \(_RM\) is Gorenstein flat. Then we prove a ``Gorenstein version'' of Nakayama and Tsuzuku's theorem on transfer of flat dimensions along Frobenius extensions: if \(_AM\) has finite Gorenstein flat dimension, then \({\mathrm{Gf}}{{\mathrm{d}}_A}\left(M \right) = {\mathrm{Gf}}{{\mathrm{d}}_R}\left(M \right)\). Moreover, it is proved that if \(R \subset S\) is a separable Frobenius extension, then for any \(A\)-module (not necessarily of finite Gorenstein flat dimension), its Gorenstein flat dimension is invariant along such ring extension.Coribbon forms on monoidal Hom-Hopf algebras.https://zbmath.org/1449.160632021-01-08T12:24:00+00:00"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.29"Zhang, Xiaohui"https://zbmath.org/authors/?q=ai:zhang.xiaohuiSummary: We define and discuss the category of generalized Hom-comodules of a monoidal Hom-Hopf algebra. We use the Tannaka dual method to describe the rigid and balanced structures. Furthermore, we give equivalent conditions of the ribbon category, and introduce the definition of the coribbon forms.Weak simple projective semimodules.https://zbmath.org/1449.160112021-01-08T12:24:00+00:00"Zeng, Huiping"https://zbmath.org/authors/?q=ai:zeng.huiping"Zhang, Chuanmei"https://zbmath.org/authors/?q=ai:zhang.chuanmei"Liu, Haoguang"https://zbmath.org/authors/?q=ai:liu.haoguangSummary: The concept of weak simple projective semimodules is introduced. Some equivalent conditions and characteristics of weak simple projective semimodules are given, homological properties are obtained. It is shown that if \(P\) is a \(wk\)-regular semimodule, then \(P\) is weak simple projective semimodules if and only if \(P\) is projective. The notion of weak quasi simple projective semi-modules is defined. Two equivalent conditions for weak quasi simple projective semimodules on a cancellative completely subtractive semiring \(R\) are given.(\(m,n\))-projective modules and (\(m,n\))-hereditary rings.https://zbmath.org/1449.160262021-01-08T12:24:00+00:00"Yang, Lulu"https://zbmath.org/authors/?q=ai:yang.lulu"Zhao, Renyu"https://zbmath.org/authors/?q=ai:zhao.renyuSummary: Let \(m, n\) be two arbitrary positive integers. By introducing the concept of (\(m, n\))-hereditary rings, and using the exact method of functor, we give some characterizations of (\(m, n\))-projective modules and (\(m, n\))-hereditary rings.\(sk\)-projective semimodules.https://zbmath.org/1449.160102021-01-08T12:24:00+00:00"Zeng, Huiping"https://zbmath.org/authors/?q=ai:zeng.huiping"Zhang, Chuanmei"https://zbmath.org/authors/?q=ai:zhang.chuanmeiSummary: In this paper the concept of \(sk\)-projective semimodules is introduced. We investigate the characterizations and properties of \(sk\)-projective semimodules, and obtain homological properties. The Schanuel lemma of \(sk\)-projective semimodules is proved. Moreover, we generalize some properties of the projective modules.On the quotient category of the module category of a finite group and its equivalence.https://zbmath.org/1449.160152021-01-08T12:24:00+00:00"Huang, Wenlin"https://zbmath.org/authors/?q=ai:huang.wenlinSummary: In this paper, we find that the class of the homomorphisms which can be decomposed by a \(p\)-divisible module is an ideal of the module category of a finite group. We construct the quotient category of this module category, analyze the zero objects and obtain three equivalence functors of this quotient category.Gorenstein FP-projective modules and its stability.https://zbmath.org/1449.160332021-01-08T12:24:00+00:00"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3"Zhao, Renyu"https://zbmath.org/authors/?q=ai:zhao.renyuSummary: The concept of Gorenstein FP-projective modules is introduced as a generalization of FP-projective modules. Some properties and equivalent characterizations of Gorenstein FP-projective modules are given. The stability of the class of Gorenstein FP-projective modules is studied.Ding projective modules over Frobenius extensions.https://zbmath.org/1449.160212021-01-08T12:24:00+00:00"Wang, Zhanping"https://zbmath.org/authors/?q=ai:wang.zhanping"Zhang, Ruijie"https://zbmath.org/authors/?q=ai:zhang.ruijieSummary: Ding projective modules over Frobenius extensions and Ding projective dimensions are investigated. Let \(R \subset A\) be a separable Frobenius extension, and let \(M\) be any left \(A\)-module. It is proved that \(M\) is a Ding projective left \(A\)-module if and only if \(M\) is Ding projective left \(R\)-module if and only if \(A{\otimes_R}M\) and \({\mathrm{Hom}}_R (A, M)\) are Ding projective left A-modules. Some conclusions on Ding projective dimensions are obtained.Some extensions of \(T\)-nilpotent rings.https://zbmath.org/1449.160472021-01-08T12:24:00+00:00"Ma, Guanglin"https://zbmath.org/authors/?q=ai:ma.guanglin"Wang, Yao"https://zbmath.org/authors/?q=ai:wang.yao"Ren, Yanli"https://zbmath.org/authors/?q=ai:ren.yanliSummary: Some extension properties of \(T\)-nilpotent rings are investigated. It is mainly proved that (1) if \(R\) is a ring and \(\alpha\) is an automorphism of \(R\), then \(R\) is a left \(T\)-nilpotent ring if and only if the skew polynomial ring \(R[x; \alpha]\) over \(R\) is a left \(T\)-nilpotent ring, if and only if the skew Laurent polynomial ring \(R[x, x^{-1}; \alpha]\) is a left \(T\)-nilpotent ring; (2) \(R\) is a left \(T\)-nilpotent ring if and only if the Nagata extension over \(R\) is a left \(T\)-nilpotent ring, if and only if the skew triangular matrix ring over \(R\) is a left \(T\)-nilpotent ring.Nonlinear Jordan higher derivable maps on triangular algebras by Lie product square zero elements.https://zbmath.org/1449.160802021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.lei"Zhu, Guowei"https://zbmath.org/authors/?q=ai:zhu.guoweiSummary: Let \(\mathcal{U}\) be a 2-torsion free triangular algebra, \(D = \{d_n\}_{n \in \textbf{N}}\) is a nonlinear Jordan higher derivable map on triangular algebra \(\mathcal{U}\) by Lie product square zero elements. In this paper, it is shown that every nonlinear Jordan higher derivable map on triangular algebra \(\mathcal{U}\) by Lie product square zero elements is a higher derivation. As its application, we get that every nonlinear Jordan higher derivable map on a nest algebra or a 2-torsion free block upper triangular matrix algebra \(\mathcal{U}\) by Lie product square zero elements is a higher derivation.Linear maps on triangular algebras for which the space of all inner derivations is Lie invariant.https://zbmath.org/1449.160822021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: Let \(\mathcal{U}\) be a triangular algebra with \({\pi_\mathcal{A}}\left({Z\left(\mathcal{U} \right)} \right) = Z\left(\mathcal{A} \right)\) and \({\pi_\mathcal{B}}\left({Z\left(\mathcal{U} \right)} \right) = Z\left(\mathcal{B} \right)\), \(\varphi\) be a \(\mathcal{R}\)-linear mapping from \(\mathcal{U}\) into itself. If \({\mathrm{ID}}\left(\mathcal{U} \right)\) is a Lie invariant subspace for \(\varphi\), then there exists a Lie derivation \(\delta\) on \(\mathcal{U}\) and a central element \(\lambda\) such that \(\varphi \left(x \right) = \delta \left(x \right) + \lambda x\) for all \(x \in \mathcal{U}\).Some results on quasi-Frobenius rings.https://zbmath.org/1449.160372021-01-08T12:24:00+00:00"Zhu, Zhanmin"https://zbmath.org/authors/?q=ai:zhu.zhanminSummary: We give some new characterizations of quasi-Frobenius rings by the generalized injectivity of rings. Some characterizations give affirmative answers to some open questions about quasi-Frobenius rings, and some characterizations improve some results on quasi-Frobenius rings.On a graph of homogenous submodules of graded modules.https://zbmath.org/1449.051352021-01-08T12:24:00+00:00"Roshan-Shekalgourabi, H."https://zbmath.org/authors/?q=ai:roshan-shekalgourabi.hajar"Hassanzadeh-Lelekaami, D."https://zbmath.org/authors/?q=ai:hassanzadeh-lelekaami.dawoudSummary: In this paper, we introduce a graph associated to a graded module over a graded ring and study the relationship between the algebraic properties of these modules and their associated graphs. In particular, the modules whose associated graph is complete, complete bipartite or star are studied and several characterizations are given.Tate homology of modules of finite Gorenstein flat dimension with respect to a semidualizing module.https://zbmath.org/1449.160222021-01-08T12:24:00+00:00"Pan, Xiaoling"https://zbmath.org/authors/?q=ai:pan.xiaoling"Liang, Li"https://zbmath.org/authors/?q=ai:liang.li.1|liang.liSummary: For a semidualing module \(C\), the Tate homology \(\widehat{\mathrm{Tor}}^{\mathcal F_C}\) of modules admitting Tate \({\mathcal F_C}\)-resolutions is investigated. In particular, a long exact sequence connecting \(\widehat{\mathrm{Tor}}^{\mathcal F_C}\), \(\mathrm{Tor}^{\mathcal F_C}\) and \(\widehat{\mathrm{Tor}}^{\mathcal{GF}_C}\) is built. As applications, the vanishing and the balance of this Tate homology theory are proved.Graded absolutely clean modules and graded level modules.https://zbmath.org/1449.160882021-01-08T12:24:00+00:00"Di, Zhenxin"https://zbmath.org/authors/?q=ai:di.zhenxin"Di, Rongrong"https://zbmath.org/authors/?q=ai:di.rongrongSummary: The concepts of graded absolutely clean modules and graded level modules over a graded ring \(R\) are introduced. It is proved that in the category of all graded \(R\)-modules, the pairs (gr-\(F_\infty\), gr-\(F{I_\infty}\)) and (gr-\(F{I_\infty}\), gr-\(F_\infty\)) form two duality pairs with some special properties, where gr-\(F{I_\infty}\) (resp., gr-\(F_\infty\)) denotes the subcategory of all graded absolutely clean \(R\)-modules (resp., graded level \(R\)-modules). As applications, it is proved that any graded left \(R\)-module has a graded absolutely clean cover and a graded level preenvelope.Central reflexive rings with an involution.https://zbmath.org/1449.160752021-01-08T12:24:00+00:00"Gao, Beilei"https://zbmath.org/authors/?q=ai:gao.beilei"Wang, Gaixia"https://zbmath.org/authors/?q=ai:wang.gaixiaSummary: We study the central reflexive properties of rings with an involution. The concept of central \(*\)-reflexive rings is introduced and investigated. It is shown that central \(*\)-reflexive rings are a generalization of reflexive rings, central reflexive rings and \(*\)-reflexive rings. Some characterizations of this class of rings are given. The related ring extensions including trivial extension, Dorroh extension and polynomial extensions are also studied.Quillen equivalence of singular model categories.https://zbmath.org/1449.180072021-01-08T12:24:00+00:00"Ren, Wei"https://zbmath.org/authors/?q=ai:ren.wei.3|ren.wei.2|ren.wei.1|ren.wei.5|ren.wei.4Summary: Let \(R\) be a left-Gorenstein ring. We construct a Quillen equivalence between singular contraderived model category and singular coderived model category. As an application, we explicitly give an equivalence \({\text bf{K}}_{\mathrm{ex}} (\mathcal{P}) \simeq {\text bf{K}}_{\mathrm{ex}} (\mathcal{I})\) for the homotopy categories of exact complexes of projective and injective modules.Quasipolar properties of a class of \(3 \times 3\) matrix rings.https://zbmath.org/1449.160552021-01-08T12:24:00+00:00"He, Yaru"https://zbmath.org/authors/?q=ai:he.yaru"Chen, Huanyin"https://zbmath.org/authors/?q=ai:chen.huanyinSummary: An element \(a \in R\) is called quasipolar if there exists \(p \in R\) such that \({p^2} = p \in {\mathrm{comm}^2} (a)\), \(a + p \in U (R)\) and \(ap \in {R^{\mathrm{qnil}}}\). A ring \(R\) is quasipolar in the case that every element in \(R\) is quasipolar. In this paper, it is determined that the \(3 \times 3\) matrix ring on the local ring \(R\) with endomorphism \(\sigma\) is quasipolar. Let \(R\) be a local ring, and let \(\sigma: R\mapsto R\) be an endomorphism of \(R\). If \(\sigma (J (R)) \subseteq J (R)\), it is proved that \({\mathcal{T}_3} (R,\sigma)\) is quasipolar if and only if \(R\) is uniquely bleached.RS rings and their applications.https://zbmath.org/1449.160712021-01-08T12:24:00+00:00"Wu, Cang"https://zbmath.org/authors/?q=ai:wu.cang"Zhao, Liang"https://zbmath.org/authors/?q=ai:zhao.liang.1By an RS ring, the authors mean an associative ring \(R\) with unity, in which every regular element is strongly regular. The main results of this paper are the following: (i) The class of RS rings properly contains the class of nilpotent-closed rings. (ii) Necessary and sufficient conditions for IC rings and von Neumann regular rings to be RS rings are given. (iii) \(R\) is abelian if, and only if, \(R\) is RS and the set of all regular elements of \(R\) is multiplicatively closed. (iv) \(R\) is reduced if, and only if, \(R\) is RS and every \(\pi\)-regular element is regular in \(R\).
Reviewer: Tibor Juhász (Eger)A bound for the rank-one transient of inhomogeneous matrix products in special case.https://zbmath.org/1449.150642021-01-08T12:24:00+00:00"Kennedy-Cochran-Patrick, Arthur"https://zbmath.org/authors/?q=ai:kennedy-cochran-patrick.arthur"Sergeev, Sergeĭ"https://zbmath.org/authors/?q=ai:sergeev.sergei-alekseevich|sergeev.sergei-m|sergeev.sergei-n"Berežný, Štefan"https://zbmath.org/authors/?q=ai:berezny.stefanSummary: We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [\textit{L. Shue, B. D. O. Anderson}, and \textit{S. Dey}, ``On steady-state properties of certain max-plus products'', in: Proceedings of the 1998 American Control Conference, Philadelphia, Pensylvania, June 24-26,1998. Piscataway, NJ: IEEE. 1909--1913 (1998; \url{doi:10.1109/acc.1998.707354}]. We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.The construction of infinitesimal Hopf algebra on the Sweedler 4-dimensional algebra.https://zbmath.org/1449.160612021-01-08T12:24:00+00:00"Liu, Renyuan"https://zbmath.org/authors/?q=ai:liu.renyuan"Zheng, Huihui"https://zbmath.org/authors/?q=ai:zheng.huihui"Yan, Jialing"https://zbmath.org/authors/?q=ai:yan.jialing"Zhang, Liangyun"https://zbmath.org/authors/?q=ai:zhang.liangyunSummary: In this paper, we mainly construct an infinitesimal Hopf algebra and its quasi-triangle Hopf algebra from the Sweedler 4-dimensional algebra and its subalgebras.The commuting graphs of finite rings.https://zbmath.org/1449.160452021-01-08T12:24:00+00:00"Dolžan, David"https://zbmath.org/authors/?q=ai:dolzan.davidSummary: In this paper, we investigate connectivity and diameters of commuting graphs of finite rings. In case of a directly decomposable ring, we calculate the diameter depending on the diameters of commuting graphs of direct summands. If the ring is indecomposable, we examine the connectedness of the commuting graph according to the number of isomorphic minimal idempotents.On absolutely FP-neat modules.https://zbmath.org/1449.160132021-01-08T12:24:00+00:00"Huang, Linna"https://zbmath.org/authors/?q=ai:huang.linna"Zhou, Dexu"https://zbmath.org/authors/?q=ai:zhou.dexuSummary: As a generalization of absolutely pure modules and absolutely neat modules, this paper uses finite simple modules to introduce absolutely FP-neat modules, gives some equivalent characterizations of absolutely FP-neat modules, proves that \(R\) is left \(N\)-ring if and only if every absolutely FP-neat left \(R\)-module is absolutely neat, and obtains some equivalent conditions under which every absolutely FP-neat left \(R\)-module is absolutely pure.Characterization of commuting weakly additive maps on a class of algebras.https://zbmath.org/1449.160692021-01-08T12:24:00+00:00"Huo, Donghua"https://zbmath.org/authors/?q=ai:huo.donghuaSummary: Let \(\mathcal{A}\) be an algebra with unit 1. A map \(f:\mathcal{A}\to \mathcal{A}\) is a weakly additive map if for every \(x, y\in\mathcal{A}\) there exist \(t_{x, y}, s_{x, y}\in\mathbb{F}\) such that \(f (x+y)=t_{x, y}f (x)+s_{x, y}f (y)\). We prove that under some conditions, if \(f\) is a commuting map, then there exist \(\lambda_0 (x)\in \mathcal{A}\) and a map \(\lambda_1\) from \(\mathcal{A}\) into \(Z (\mathcal{A})\) such that \(f (x)=\lambda_0 (x)x+\lambda_1 (x)\) for all \(x\in \mathcal{A}\). As an application, a class of commuting weakly additive maps on \(M_n (\mathbb{F})\) are characterized.On the dual of weakly prime and semiprime modules.https://zbmath.org/1449.160022021-01-08T12:24:00+00:00"Beyranvand, Reza"https://zbmath.org/authors/?q=ai:beyranvand.rezaSummary: The weakly second modules (the dual of weakly prime modules) were introduced in [the author and \textit{F. Rastgoo}, Hacet. J. Math. Stat. 45, No. 5, 1355--1366 (2016; Zbl 1369.16002)]. In this paper we introduce and study the semisecond and strongly second modules. Let \(R\) be a ring and \(M\) be an \(R\)-module. We show that \(M\) is semisecond if and only if \(MI=MI^2\) for any ideal \(I\) of \(R\). It is shown that every sum of the second submodules of \(M\) is a semisecond submodule of \(M\). Also if \(M\) is an Artinian module, then \(M\) has only a finite number of maximal semisecond submodules. We prove that every strongly second submodule of \(M\) is second and every minimal submodule of \(M\) is strongly second. If every nonzero submodule of \(M\) is (weakly) second, then \(M\) is called fully (weakly) second. It is shown that if \(R\) is a commutative ring, then \(M\) is fully second if and only if \(M\) is fully weakly second, if and only if \(M\) is a homogeneous semisimple module.Some new characterizations on EP elements.https://zbmath.org/1449.160722021-01-08T12:24:00+00:00"Shi, Liyan"https://zbmath.org/authors/?q=ai:shi.liyan"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li|ma.li.1"Wei, Junchao"https://zbmath.org/authors/?q=ai:wei.junchaoSummary: In this paper, we give some conditions for a core invertible element being EP element in a \(*\)-ring \(R\). We mainly prove the following results by discussing the solutions of some given equations: Let \(a \in {R^\#} \cap {R^\dag}\). Then \(a \in {R^{{\mathrm{EP}}}}\) if and only if one of the following equations has at least one solution in \({\chi_a}\), where \({\chi_a} = \{a, a^*, a^\dag, a^\#, (a^\#)^*, (a^\dag)^*\}\): (1) \(xa{a^*}a = {a^*}{a^2}x\); (2)\({a^*}a{a^*}x = x{a^*}{a^*}a\); (3) \(x{a^*}a{a^*} = a{a^*}{a^*}x\); (4) \(a{a^*}ax = x{a^2}{a^*}\).Irreducible \& strongly irreducible bi-ideals of \(\Gamma\)-so-rings.https://zbmath.org/1449.161032021-01-08T12:24:00+00:00"Srinivasa Rao, P. V."https://zbmath.org/authors/?q=ai:rao.p-v-srinivasa"Siva Mala, M."https://zbmath.org/authors/?q=ai:mala.m-sivaSummary: The set of all partial functions over a set under a natural addition (disjoint-domain sum), functional composition and functional relation on them, forms a \(\Gamma\)-so-ring. In this paper we introduce the notions of irreducible bi-ideal, strongly irreducible bi-ideal and strongly prime bi-ideals of \(\Gamma\)-so-rings and we prove that a bi-ideal is strongly irreducible if and only if it is strongly prime in a class of \(\Gamma\)-so-rings.Three dimensional semigroup algebras.https://zbmath.org/1449.200552021-01-08T12:24:00+00:00"Ji, Yingdan"https://zbmath.org/authors/?q=ai:ji.yingdan"Luo, Yanfeng"https://zbmath.org/authors/?q=ai:luo.yan-fengSummary: In this paper, the authors provide the idempotent sets and Jacobson radicals of all three dimensional semigroup algebras, and then classify these algebras up to isomorphism, where the results depend on the characteristic of the ground algebraically closed field. Then the representation type of these semigroup algebras are investigated by computing certain quivers. The authors also prove that a three (resp., two) dimensional semigroup algebra is cellular if and only if it is commutative. As a by-product, it is shown that the semigroup algebra of a left zero band is cellular if and only if it is a semilattice.Gröbner-Shirshov basis of non-degenerate affine Hecke algebras of type \({A_n}\).https://zbmath.org/1449.161042021-01-08T12:24:00+00:00"Munayim, Dilxat"https://zbmath.org/authors/?q=ai:munayim.dilxat"Abdukadir, Obul"https://zbmath.org/authors/?q=ai:abdukadir.obulSummary: In this paper, by computing the compositions, we give a Gröbner-Shirshov basis of non-degenerate affine Hecke algebra of type \({A_n}\). By using this Gröbner-Shirshov basis and the composition-diamond lemma of associative algebras, we give a linear basis of the non-degenerate affine Hecke algebra of type \({A_n}\).Cycles in monomial algebras.https://zbmath.org/1449.160202021-01-08T12:24:00+00:00"Shi, Hongbo"https://zbmath.org/authors/?q=ai:shi.hongboSummary: The objective of this note is to use the combinatorial algorithm topdown to show some properties of cycles in a monomial algebra, from which some well-known results are deduced immediately, providing very short combinatorial arguments to them.On the structures of \({K_2} (\mathbb{Z}[{C_6}])\) and \({K_2} (\mathbb{Z}[{C_{10}}])\).https://zbmath.org/1449.190012021-01-08T12:24:00+00:00"Zhang, Yakun"https://zbmath.org/authors/?q=ai:zhang.yakun"Tang, Guoping"https://zbmath.org/authors/?q=ai:tang.guopingSummary: Let \({C_n}\) be a cyclic group of order \(n\). We obtain the explicit structures of \({K_2} (\mathbb{Z}[{C_6}])\) and \(W{h_2} ({C_6})\), and the 2-primary torsion subgroup of \({K_2} (\mathbb{Z}[{C_{10}}])\) and \(W{h_2} ({C_{10}})\). Besides, we give the explicit structures of \({K_2} (\mathbb{Z}[{\zeta_3}][{C_2}])\) and the 2-primary torsion subgroup of \({K_2} (\mathbb{Z}[{\zeta_5}][{C_2}])\).Symmetry of extending properties in nonsingular Utumi rings.https://zbmath.org/1449.160282021-01-08T12:24:00+00:00"Do, Thuat"https://zbmath.org/authors/?q=ai:do.thuat-van"Hoang, Hai Dinh"https://zbmath.org/authors/?q=ai:hoang.hai-dinh"Tu, Truong Dinh"https://zbmath.org/authors/?q=ai:tu.truong-dinhSummary: This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp., right min, right max-min) CS if and only if it is left min (resp., left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.Armendariz ring with weakly semicommutativity.https://zbmath.org/1449.160512021-01-08T12:24:00+00:00"Singh, Sushma"https://zbmath.org/authors/?q=ai:singh.sushma"Prakash, Om"https://zbmath.org/authors/?q=ai:prakash.omSummary: In this article, we introduce the weak ideal-Armendariz ring which combines Armendariz ring and weakly semicommutative ring. In fact, it is a generalization of an ideal-Armendariz ring. We investigate some properties of weak ideal-Armendariz rings and prove that \(R\) is a weak ideal-Armendariz ring if and only if \(R[x]\) is a weak ideal-Armendariz ring. Also, we generalize weak ideal-Armendariz as strongly nil-IFP and some properties are discussed which distinguishes it from other existing structures. We prove that if \(I\) is a semicommutative ideal of a ring \(R\) and \(\frac{R}{I}\) is strongly nil-IFP, then \(R\) is strongly nil-IFP. Moreover, if \(R\) is 2-primal, then \(R[x]/\langle {x^n}\rangle\) is strongly nil-IFP.Properties of soft exact sequences.https://zbmath.org/1449.160052021-01-08T12:24:00+00:00"Wu, Yue"https://zbmath.org/authors/?q=ai:wu.yue"Ma, Jing"https://zbmath.org/authors/?q=ai:ma.jingSummary: By using module theory and the basic properties of soft modules, we discuss the decomposition properties of soft homomorphisms. Firstly, we define a single soft homomorphism of soft modules and a soft exact sequence. Secondly, we prove that every soft homomorphism can be decomposed into the composition of an epimorphism and a monomorphism. Finally, we discuss the basic properties of soft exact sequences, and give equivalent conditions for several kinds of simple soft modulus sequence to be exact, and construct a new soft exact sequence by using two soft exact sequences.Ideals, centers, and generalized centers of nearrings of functions determined by a single invariant subgroup.https://zbmath.org/1449.160922021-01-08T12:24:00+00:00"Alan, Cannon G."https://zbmath.org/authors/?q=ai:alan.cannon-g"Secmen, Gulendam Aysu Bilgin"https://zbmath.org/authors/?q=ai:secmen.gulendam-aysu-bilginSummary: Let \(G\) be a finite group and let \(H\) be a nonzero, proper subgroup of \(G\). Then \(N = \{f:G \to G|f (0) = 0\; {\mathrm{and}}\; f (H)\subseteq H\}\) is a nearring under pointwise addition and function composition. We determine all ideals of \(N\), the center of \(N\), and the generalized center of \(N\), and find necessary and sufficient conditions for the center to be a subnearring of \(N\).Skew polynomial extensions of semi-Baer and semi-quasi Baer rings.https://zbmath.org/1449.160502021-01-08T12:24:00+00:00"Patil, Avinash"https://zbmath.org/authors/?q=ai:patil.avinash-aSummary: Let \(R\) be a ring and \(\alpha \) be an endomorphism of \(R\). In this paper, we introduce the concepts of \(\alpha \)-semi-Baer, \(\alpha \)-semi-quasi Baer rings as a generalization of semi-Baer and semi-quasi Baer rings respectively. We investigate the interrelation between \(\alpha \)-semi-Baer and \(\alpha \)-semi-quasi-Baer properties of \(R\) and the skew polynomial ring \(R[x; \alpha]\). Examples are provided to delimit the results.Abian's relation on semirings.https://zbmath.org/1449.160962021-01-08T12:24:00+00:00"Khatun, Sarifa"https://zbmath.org/authors/?q=ai:khatun.sarifa"Sircar, Jayasri"https://zbmath.org/authors/?q=ai:sircar.jayasri"Abu Nayeem, Sk. Md."https://zbmath.org/authors/?q=ai:abu-nayeem.sk-mdSummary: In this paper, we introduce a relation called Abian's relation on a semiring. A semiring endowed with Abian's order is called an Abian's semiring. Here we obtain a sufficient condition for an Abian's semiring to be a partially ordered semiring with respect to Abian's relation. Also we study different properties of the comparability graph of an Abian's semiring.Modules whose \(h\)-closed submodules are direct summands.https://zbmath.org/1449.160032021-01-08T12:24:00+00:00"Kara, Yeliz"https://zbmath.org/authors/?q=ai:kara.yelizSummary: This paper is based on the class of modules whose \(h\)-closed submodules are direct summands. We introduce and investigate the structural properties for the former class of modules and we elaborate our results with lifting homomorphisms.Results in prime rings with generalized two sided \(\alpha\)-derivations.https://zbmath.org/1449.160402021-01-08T12:24:00+00:00"Boua, Abdelkarim"https://zbmath.org/authors/?q=ai:boua.abdelkarimSummary: In the present paper, we study the commutativity of prime rings satisfying certain differential identities involving generalized two sided \(\alpha\)-derivations on rings. Furthermore, we give examples to show that the restrictions imposed on the hypothesis of various theorems are not superfluous.A note on Noetherian modules.https://zbmath.org/1449.160462021-01-08T12:24:00+00:00"Soontharanon, Jarunee"https://zbmath.org/authors/?q=ai:soontharanon.jarunee"Nguyen, Nghiem Dang Hoa"https://zbmath.org/authors/?q=ai:nguyen.nghiem-dang-hoa"Nguyen Van Sanh"https://zbmath.org/authors/?q=ai:nguyen-van-sanh.Summary: In this note, we introduce a class of nearly prime submodules and prove that a finitely generated right \(R\)-module \(M\) is Noetherian if and only if every nearly prime submodule is finitely generated.Stability of Gorenstein classes relative to Wakamatsu tilting modules.https://zbmath.org/1449.160292021-01-08T12:24:00+00:00"He, Donglin"https://zbmath.org/authors/?q=ai:he.donglin"Li, Yuyan"https://zbmath.org/authors/?q=ai:li.yuyanSummary: Using homological algebra methods, the paper investigates the stability of Gorenstein classes relative to Wakamatsu tilting module, and proves that \( (g{i_\omega})^{n+1} = (g{i_\omega})^n\) for any integer \(n \ge 1\).BiHom-\(H\)-pseudoalgebras and their constructions.https://zbmath.org/1449.160682021-01-08T12:24:00+00:00"Shi, Guodong"https://zbmath.org/authors/?q=ai:shi.guodong"Wang, Shuanhong"https://zbmath.org/authors/?q=ai:wang.shuanhongSummary: The definition and an example of BiHom-associative \(H\)-pseudoalgebra are given. A BiHom-\(H\)-pseudoalgebra is an \(H\)-pseudoalgebra \( (A, \mu)\) with two maps \(\alpha, \beta \in {\mathrm{Hom}}_H (A, A)\) satisfying the BiHom-associative law which generalizes BiHom-associative algebras and associative \(H\)-pseudoalgebras. Secondly, a method which is called the Yau twist of constructing BiHom-associative \(H\)-pseudoaglebra \( (A, ({I_{H \otimes H}}{\otimes_H}\alpha)\mu, \alpha, \beta)\) from an associative \(H\)-pseudoalgebra \( (A, \mu)\) and two maps of \(H\)-pseudoalgebras \(\alpha, \beta\), is introduced. Thirdly, a generalized form of the Yau twist is discussed. It concerns constructing a BiHom-associative \(H\)-pseudoalgebra \( (A, \mu (\alpha \otimes \beta), {\alpha^\sim}\alpha, {\beta^\sim}\beta)\) from a BiHom-associative \(H\)-pseudoalgebra \( (A, \mu, {\alpha^\sim}, {\beta^\sim})\) and two maps \(\alpha, \beta \in {\mathrm{Hom}}_H (A, A)\). Finally, a method of constructing BiHom-associative \(H\)-pseudoalgebra on tensor product space \(A \otimes B\) of two BiHom-associative \(H\)-pseudoalgebras is given.The properties of determinants for matrix multiplications over commutative semirings.https://zbmath.org/1449.150772021-01-08T12:24:00+00:00"Liu, Yijin"https://zbmath.org/authors/?q=ai:liu.yijin"Wang, Xueping"https://zbmath.org/authors/?q=ai:wang.xueping.1Summary: This paper mainly investigates the properties of determinants for matrix multiplications over commutative semirings. It discusses the relationships between the determinant of matrix multiplications and the multiplication of determinants for matrices, and shows the relationships between the multiplication of adjoint matrices and the adjoint matrix of matrix multiplications.The existence of \(\#\)-injective envelopes of complexes.https://zbmath.org/1449.160182021-01-08T12:24:00+00:00"Liang, Li"https://zbmath.org/authors/?q=ai:liang.li"Yang, Gang"https://zbmath.org/authors/?q=ai:yang.gangSummary: Let \(dw\widetilde{\mathcal{I}}\) denote the class of \(\#\)-injective complexes of left \(R\)-modules (i.e., complexes of injective left \(R\)-modules). We prove that over left Noetherian rings \(R\), the pair \( (^\perp(dw\widetilde{\mathcal{I}}), dw\widetilde{\mathcal{I}})\) is a perfect injective cotorsion pair. In particular, we get that every complex of left \(R\)-modules has a \(\#\)-injective envelope. As an application, we prove that over left Noetherian rings \(R\), every complex of left \(R\)-modules has a special \(\mathcal{E}_{\mathrm{tac}} (\mathcal{I})\)-preenvelope, where \(\mathcal{E}_{\mathrm{tac}} (\mathcal{I})\) is the class of complete acyclic complexes of injective left \(R\)-modules.\(GPF\)-properties of group rings.https://zbmath.org/1449.160492021-01-08T12:24:00+00:00"Odetalllah, Huda"https://zbmath.org/authors/?q=ai:odetalllah.huda"Al-Ezeh, Hasan"https://zbmath.org/authors/?q=ai:al-ezeh.hasan"Abuosba, Emad"https://zbmath.org/authors/?q=ai:abuosba.emadSummary: All rings \(R\) in this article are assumed to be commutative with unity \(1\ne 0\). A ring \(R\) is called a \(GPF\)-ring if for every \(a\in R\) there exists a positive integer \(n\) such that the annihilator ideal \(\operatorname{Ann}_R (a^ n)\) is pure. We prove that for a ring \(R\) and an abelian group \(G\), if the group ring \(RG\) is a \(GPF\)-ring then so is \(R\). Moreover, if \(G\) is a finite abelian group then \(|G|\) is a unit or a zero-divisor in \(R\). We prove that if \(G\) is a group such that for every nontrivial subgroup \(H\) of \(G\), \([G:H]<\infty\), then the group ring \(RG\) is a \(GPF\)-ring if and only if \(RH\) is a \(GPF\)-ring for each finitely generated subgroup \(H\) of \(G\). It is proved that if \(R\) is a local ring and \(RG\) is a \(U\)-group ring, then \(RG\) is a \(GPF\)-ring if and only if
\(R\) is a \(GPF\)-ring and \(p\in \operatorname{Nil}(R)\). Finally, we prove that if \(R\) is a semisimple ring and \(G\) is a finite group such that \(|G|^{-1}\in R\), then \(RG\) is a \(GPF\)-ring if and only if \(RG\) is a \(PF\)-ring.Non-additive Lie centralizer of infinite strictly upper triangular matrices.https://zbmath.org/1449.160542021-01-08T12:24:00+00:00"Aiat Hadj, D. A."https://zbmath.org/authors/?q=ai:aiat-hadj.driss-ahmedSummary: Let \(\mathcal{F}\) be an field of zero characteristic and \(N_\infty(\mathcal{F})\) be the algebra of infinite strictly upper triangular matrices with entries in \(\mathcal{F}\), and \(f:N_\infty(\mathcal{F})\rightarrow N_\infty(\mathcal{F})\) be a non-additive Lie centralizer of \(N_\infty(\mathcal{F})\), that is, a map satisfying that \(f([X,Y])=[f(X),Y]\) for all \(X,Y\in N_\infty(\mathcal{F})\). We prove that \(f(X)=\lambda X\), where \(\lambda \in \mathcal{F}\).Ring endomorphisms with nil-shifting property.https://zbmath.org/1449.160532021-01-08T12:24:00+00:00"Ahmed, C. A. K."https://zbmath.org/authors/?q=ai:ahmed.chenar-abdul-kareem"Salim, R. T. M."https://zbmath.org/authors/?q=ai:salim.r-t-mSummary: \textit{P. M. Cohn} [Bull. Lond. Math. Soc. 31, No. 6, 641--648 (1999; Zbl 1021.16019)] called a ring \(R\) is reversible if whenever \(ab = 0,\) then \(ba = 0\) for \(a,b \in R\). The reversible property is an important role in noncommutative ring theory. Recently, \textit{A. M. Abdul-Jabbar} et al. [Commun. Algebra 45, No. 11, 4881--4895 (2017; Zbl 1388.16039)] studied the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring). In this paper, we extend the CNZ property of a ring as follows: Let \(R\) be a ring and \(\alpha\) an endomorphism of \(R\), we say that \(R\) is right (resp., left) \(\alpha\)-nil-shifting ring if whenever \(a\alpha(b) = 0 \) (resp., \(\alpha(a)b = 0)\) for nilpotents \(a,b\) in \(R, b\alpha(a) = 0 \) (resp., \(\alpha(b)a= 0)\). The characterization of \(\alpha\)-nil-shifting rings and their related properties are investigated.Gorenstein \(F{P_n}\)-injective and Gorenstein \(F{P_n}\)-flat modules.https://zbmath.org/1449.160062021-01-08T12:24:00+00:00"Chen, Dong"https://zbmath.org/authors/?q=ai:chen.dong"Hu, Kui"https://zbmath.org/authors/?q=ai:hu.kuiSummary: The concepts of Gorenstein \(F{P_n}\)-injective modules and Gorenstein \(F{P_n}\)-flat modules are introduced. We study the basic properties of these two classes of modules firstly and then discuss the \(F{P_n}\)-injective dimension of Gorenstein \(F{P_n}\)-injective modules and the structures of Gorenstein \(F{P_n}\)-injective modules over an \(n\)-coherent ring. Finally, the conditions under which each \(R\)-module is a Gorenstein \(F{P_n}\)-injective module are given.Belitskii's reduction and standard form of matrix pairs.https://zbmath.org/1449.150352021-01-08T12:24:00+00:00"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.1|zhang.chao.9|zhang.chao.4|zhang.chao.7|zhang.chao.2|zhang.chao.3|zhang.chao.5|zhang.chao.8|zhang.chao.6|zhang.chao"Cai, Hongyan"https://zbmath.org/authors/?q=ai:cai.hongyanSummary: In the representation theory, the algebra \(\Gamma = k\langle {x, y} \rangle\) plays an important role in the research of the representation type of algebras. In this paper, all the representations of \(\Gamma\) up to isomorphisms are described by using the Belitskii's reduction. Equivalently, we determine the standard form of matrix pairs of size three. As an application, we obtain the number of parameters of this linear matrix problem based on the standard form.The primary congruence and the primary decomposition of congruence on semirings.https://zbmath.org/1449.161002021-01-08T12:24:00+00:00"Wu, Ya'nan"https://zbmath.org/authors/?q=ai:wu.yanan"Ren, Miaomiao"https://zbmath.org/authors/?q=ai:ren.miaomiaoSummary: We study primary congruences on commutative idempotent semirings, give definition of \(\rho\)-congruence, and obtain some results of their structures. On this basis, a uniqueness theorem is obtained by studying the minimal primary congruence decomposition.Difference bases in finite abelian groups.https://zbmath.org/1449.050342021-01-08T12:24:00+00:00"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Gavrylkiv, Volodymyr"https://zbmath.org/authors/?q=ai:gavrylkiv.volodymyr-mSummary: A subset \(B\) of a group \(G\) is called a difference basis of \(G\) if each element \(g\in G\) can be written as the difference \(g =ab^{-1}\) of some elements \(a,b\in B\). The smallest cardinality \(|B|\) of a difference basis \(B\subset G\) is called the difference size of \(G\) and is denoted by \(\Delta [G]\). The fraction \(\partial [G]:=\Delta [G]/\sqrt{|G|}\) is called the difference characteristic of \(G\). Using properties of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite abelian groups. In particular, we prove that for a prime number \(p\geq 11\), any finite abelian \(p\)-group \(G\) has difference characteristic \(\partial [G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot \sup \partial [C_{p^k}]<\sqrt{2}\cdot \frac{\sqrt{p}-1}{\sqrt{p}-3}\). Also we calculate the difference sizes of all abelian groups of cardinality less than 96.Note on the relations of the positive/negative determinants \(|I + XY|\) over commutative semirings.https://zbmath.org/1449.150112021-01-08T12:24:00+00:00"Liu, Yijin"https://zbmath.org/authors/?q=ai:liu.yijin"Wu, Li"https://zbmath.org/authors/?q=ai:wu.li"Wang, Xueping"https://zbmath.org/authors/?q=ai:wang.xueping.1Summary: In this paper, the relations of the positive/negative determinants \(|I + XY|\) over commutative semirings are investigated.On Lie ideals and symmetric generalized \((\alpha, \beta)\)-biderivation in a prime ring.https://zbmath.org/1449.160432021-01-08T12:24:00+00:00"Rehman, Nadeem Ur"https://zbmath.org/authors/?q=ai:rehman.nadeem-ur"Huang, Shuliang"https://zbmath.org/authors/?q=ai:huang.shuliangSummary: Let \(\mathfrak{R}\) be a prime ring with char\((\mathfrak{R}) \neq 2\). A biadditive symmetric map \(\Delta : \mathfrak{R} \times \mathfrak{R} \to \mathfrak{R}\) is called symmetric \((\alpha, \beta)\)-biderivation if, for any fixed \(y \in \mathfrak{R}\), the map \(x \mapsto \Delta(x, y)\) is a \((\alpha, \beta)\)-derivation. A symmetric biadditive map \(\Gamma : \mathfrak{R} \times \mathfrak{R} \to \mathfrak{R}\) is a symmetric generalized \((\alpha, \beta)\)-biderivation if for any fixed \(y \in \mathfrak{R}\), the map \(x\mapsto \Gamma(x, y)\) is a generalized \((\alpha, \beta)\)-derivation of \(\mathfrak{R}\) associated with the \((\alpha, \beta)\)-derivation \(\Delta(., y)\). In the present paper, we investigate the commutativity of a ring having a generalized \((\alpha, \beta)\)-biderivation satisfying certain algebraic conditions.Graded \(w\)-modules over graded rings.https://zbmath.org/1449.160892021-01-08T12:24:00+00:00"Wu, Xiaoying"https://zbmath.org/authors/?q=ai:wu.xiaoying"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Liang, Chunmei"https://zbmath.org/authors/?q=ai:liang.chunmeiSummary: In this paper, \(R=\bigoplus\limits_{\sigma\in G}{R_\sigma}\) is a commutative \(G\)-graded ring with identity 1. We also call \(R\) a graded ring for short. Besides, graded \(w\)-modules and other related conceptions over a graded ring \(R\) are introduced. It is shown that: (1) let \(J\) be a finitely generated graded ideal of \(R\). Then \(J\) is a graded GV-ideal if and only if \(J\) is a GV-ideal. (2) If \(M\) is a graded GV-torsion-free module (respectively, GV-torsion module), then the \(\sigma\)-suspended graded module \(M (\sigma)\) is also a graded GV-torsion-free module (respectively, GV-torsion module). (3) Let \(M\) be a graded \(w\)-module and \(N\) be a graded submodule of \(M\). Then \(N\) is a graded \(w\)-module if and only if \(N\) is a \(w\)-module. Especially, a graded \(w\)-ideal of \(R\) is a \(w\)-ideal.Covering and enveloping on \(w\)-operation.https://zbmath.org/1449.160122021-01-08T12:24:00+00:00"Zhang, Xiaolei"https://zbmath.org/authors/?q=ai:zhang.xiaoleiSummary: In this paper, we study the covering and enveloping properties of classes of \(w\)-modules and \(w\)-flat modules. If all the GV-ideals over a commutative ring are finitely presented, then the class of \(w\)-modules is proved to be covering. We prove that for any ring, the class of \(w\)-flat modules is covering and the class of \(w\)-modules is enveloping. Lastly, we prove that the class of \(w\)-flat modules is preenveloping if and only if it is closed under direct product.F-Gorenstein flat dimension.https://zbmath.org/1449.160082021-01-08T12:24:00+00:00"Liu, Miao"https://zbmath.org/authors/?q=ai:liu.miao"Yang, Xiaoyan"https://zbmath.org/authors/?q=ai:yang.xiaoyanSummary: Denote the class of F-Gorenstein flat \(R\)-modules by \(\mathcal{H} (\mathcal{F})\). Some descriptions of F-Gorenstein flat dimension are given for an arbitrary ring. As an application, it is proved that if the F-Gorenstein flat dimension of all \(R\)-modules is not more than 1, then the pair \( (\mathcal{H} (\mathcal{F}), \mathcal{H} (\mathcal{F})^\bot)\) forms a complete and hereditary cotorsion pair.Additivity of biderivable maps on generalized matrix algebras.https://zbmath.org/1449.160782021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: Let \(\mathcal{G}\) be a generalized matrix algebra, \(\varphi:\mathcal{G} \times \mathcal{G} \to \mathcal{G}\) be a mapping of \(\mathcal{G}\) (without assumption of additivity on each argument). If \(\varphi\) satisfies \(\varphi (XY, Z) = \varphi (X, Z)Y + X\varphi (Y, Z)\) and \(\varphi (X, YZ) = \varphi (X, Y)Z + Y\varphi (X, Z)\) for all \(X, Y, Z \in \mathcal{G}\), then \(\varphi\) is a biderivation.Higher \(\xi \)-Lie derivable maps on triangular algebras at reciprocal elements.https://zbmath.org/1449.160862021-01-08T12:24:00+00:00"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xia"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(\mathcal{U} = {\mathrm{Tri}} (\mathcal{A, M, B})\) be a triangular algebra with identity 1, \({1_\mathcal{A}}\), \({1_\mathcal{B}}\) be the unit of \(\mathcal{A}\) and \(\mathcal{B}\), respectively. For any \(A \in \mathcal{A}, B \in \mathcal{B}\), there are integers \({k_1}, {k_2}\) respectively, making \({k_1}{1_\mathcal{A}}-A, {k_2}{1_\mathcal{B}}-B\) invertible in triangular algebras. \(\{\varphi_n\}_{n \in N}: \mathcal{U} \to \mathcal{U}\) be a sequence of linear maps. In this paper, we prove that, if \(\{\varphi_n\}_{n \in N}\) satisfies \({\varphi_n} ([U, V]_\xi) = \sum\limits_{i + j = n} {\varphi_i} (U){\varphi_j} (V)-\xi{\varphi_i} (V){\varphi_j} (U)\) \((\xi \ne 0, 1)\), for any \(U, V \in \mathcal{U}\) with \(UV = VU = 1\), then \(\{\varphi_n\}_{n \in N}\) is a higher derivation, where \({\varphi_0} = \mathrm{id}_0\) is the identity map, \([U,V]_\xi = UV - \xi VU\).Coleman automorphisms of extensions of the finite groups by some groups.https://zbmath.org/1449.200262021-01-08T12:24:00+00:00"Zhao, Lele"https://zbmath.org/authors/?q=ai:zhao.lele"Hai, Jinke"https://zbmath.org/authors/?q=ai:hai.jinkeSummary: Let \(G\) be an extension of a finite characteristically simple group by a finite abelian group or a finite non-abelian simple group. It is shown that under some conditions every Coleman automorphism of \(G\) is an inner automorphism.Relative position of three subspaces in a Hilbert space.https://zbmath.org/1449.460232021-01-08T12:24:00+00:00"Enomoto, Masatoshi"https://zbmath.org/authors/?q=ai:enomoto.masatoshi"Watatani, Yasuo"https://zbmath.org/authors/?q=ai:watatani.yasuoSummary: We study the relative position of three subspaces in an infinite dimensional Hilbert space. In the finite-dimensional case over an arbitrary field, \textit{S. Brenner} [J. Algebra 6, 100--114 (1967; Zbl 0229.16020)] described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space over the complex numbers.Reversible and reflexive properties for rings with involution.https://zbmath.org/1449.160742021-01-08T12:24:00+00:00"Aburawash, Usama A."https://zbmath.org/authors/?q=ai:aburawash.usama-a"Saad, Muhammad"https://zbmath.org/authors/?q=ai:saad.muhammadSummary: In this note, we give a generalization for the class of *-IFP rings. Moreover, we introduce *-reversible and *-reflexive *-rings, which represent the involutive versions of reversible and reflexive rings and expose their properties. Nevertheless, the relation between these rings and those without involution are indicated. Moreover, a nontrivial generalization for *-reflexive *-rings is given. Finally, in *-reversible *-rings it is shown that each nilpotent element is *-nilpotent and Köthe's conjecture has a strong affirmative solution.Functional equations characterizing \(\sigma\)-derivations.https://zbmath.org/1449.160832021-01-08T12:24:00+00:00"Hosseini, Amin"https://zbmath.org/authors/?q=ai:hosseini.amin"Karizaki, Mehdi Mohammadzadeh"https://zbmath.org/authors/?q=ai:karizaki.mehdi-mohammadzadehSummary: The main purpose of this article is to prove the following result: For integers \(m, n\) with \(m \ge 0\), \(n \ge 0\), and \(m+n\ne 0\), let \(\mathcal{R}\) be an \((m+n+2)!\)-torsion free prime ring with the identity element \(e\). Suppose that \(d,\sigma: \mathcal{R} \to\mathcal{R}\) are two additive mappings such that \(\sigma\) is a monomorphism with \(\sigma(e) = e\), and \(d(\mathcal{R})\subseteq \sigma(\mathcal{R})\). If \(d\) and \(\sigma\) satisfy both of the equations \[d(xy)(\sigma(z)-z)-d(x)(\sigma(yz)-\sigma(y)z) + \sigma(xy)d(z)-\sigma(x)(d(yz)-d(y)z) = 0\] and \[d(x^{m+n+1}) = (m + n + 1)\sigma(x^m)d(x)\sigma (x^n)\] for all \(x, y, z \in \mathcal{R}\), then \(d\) is a \(\sigma\)-derivation.Globalizations for partial Hom-module coalgebras.https://zbmath.org/1449.160652021-01-08T12:24:00+00:00"Cheng, Wenjing"https://zbmath.org/authors/?q=ai:cheng.wenjing"Guo, Huaiwen"https://zbmath.org/authors/?q=ai:guo.huaiwen"Chen, Quanguo"https://zbmath.org/authors/?q=ai:chen.quanguoSummary: By introducing the concept of a partial Hom-module coalgebra, globalizations for partial Hom-module coalgebras are considered. It is proved that every partial Hom-module coalgebra has a globalization.Cotorsion theories relative to tilting pairs.https://zbmath.org/1449.160342021-01-08T12:24:00+00:00"He, Donglin"https://zbmath.org/authors/?q=ai:he.donglin"Li, Yuyan"https://zbmath.org/authors/?q=ai:li.yuyanSummary: Let \( (C,T)\) be a tilting pair over an Artin algebra \(\Lambda\). Using methods of homological algebra, precovers and preenvelopes relative to the tilting pair \( (C,T)\) are discussed. The results indicate that \( (^{\bot_1} ({T^\bot}), {T^\bot})\) is a complete hereditary cotorsion theory when mod \(\Lambda \subseteq {\mathrm{Gen}} (C)\) and \({e_\Lambda} (C, -) < \infty\), where Gen\( (C)\) denotes the class of modules generated by \(C\), \({e_\Lambda} (C, M) = {\mathrm{sup}}\{i \in N|{\mathrm{Ext}}_\Lambda^i (C, M) \ne 0\}\) and \({e_\Lambda} (C, -) = {\mathrm{sup}}\{{e_\Lambda} (C, M)|M \in {\mathrm{mod}}\;\Lambda\}\).Strong Ding projective and injective complexes.https://zbmath.org/1449.160192021-01-08T12:24:00+00:00"Zhang, Cuiping"https://zbmath.org/authors/?q=ai:zhang.cuiping"Guo, Huiying"https://zbmath.org/authors/?q=ai:guo.huiyingSummary: This paper generalizes strong Ding projective (injective) modules to strong Ding projective (injective) complexes. Further, it is proved that each degree of strong Ding projective (injective) complexes \(G\) is strong Ding projective (injective) modules, and for any flat (\(FP\)-injective) complexes \(F (J)\), \({\mathrm{Hom}}^\cdot (G,F)\) (\({\mathrm{Hom}}^\cdot (F,G))\) is exact. Ding projective (injective) complexes are direct summands of strong Ding projective (injective) complexes.\(n\)-strongly \(\mathcal{W}\)-Gorenstein graded modules.https://zbmath.org/1449.160912021-01-08T12:24:00+00:00"Zhao, Renyu"https://zbmath.org/authors/?q=ai:zhao.renyu"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3Summary: Let \(R\) be a graded ring, \(n\) be a positive integer, and \(\mathcal{W}\) be a self-orthogonal class of graded \(R\)-modules. The notion of \(n\)-strongly \(\mathcal{W}\)-Gorenstein graded modules is introduced. Some equivalent characterizations of \(n\)-strongly \(\mathcal{W}\)-Gorenstein graded modules are given and the relation between \(n\)-strongly \(\mathcal{W}\)-Gorenstein graded modules and \(m\)-strongly \(\mathcal{W}\)-Gorenstein graded modules is discussed.Foxby equivalences of Cartan-Eilenberg complexes.https://zbmath.org/1449.160162021-01-08T12:24:00+00:00"Zhang, Chunxia"https://zbmath.org/authors/?q=ai:zhang.chunxia"Que, Chunyue"https://zbmath.org/authors/?q=ai:que.chunyueSummary: Let \(R\) be a commutative Noetherian ring with a semi-dualizing module \(C\). We introduce CE (abbreviation for Cartan-Eilenberg) Auslander class CE-\({\mathcal{A}_c} (R)\) and CE Bass class CE-\({\mathcal{B}_c} (R)\), and extend the Foxby equivalence to the setting of CE complexes.Some properties of the \(\mathcal{M}_R^l (\Omega)\) category.https://zbmath.org/1449.160142021-01-08T12:24:00+00:00"Geng, Jun"https://zbmath.org/authors/?q=ai:geng.jun"Tang, Jiangang"https://zbmath.org/authors/?q=ai:tang.jiangangSummary: In this paper, we research the coequalizer and the coproduct in the category of \(\Omega\)-left-\(R\)-modules, and give the relationship of coproduct and coequalizer between the \(\Omega\)-left-\(R\)-modules category and the left-\(R\)-modules category. Furthermore, we prove that the category of \(\Omega\)-left-\(R\)-modules is a cocomplete category.Strongly Gorenstein injective modules with respect to a cotorsion pair.https://zbmath.org/1449.160322021-01-08T12:24:00+00:00"Wang, Zhanping"https://zbmath.org/authors/?q=ai:wang.zhanping"Yuan, Kaiying"https://zbmath.org/authors/?q=ai:yuan.kaiyingSummary: As a generalization of strongly Gorenstein injective modules, this paper introduces and studies strongly Gorenstein injective modules with respect to a complete and hereditary cotorsion pair \( (\mathcal{X},\mathcal{Y})\), that is, strongly Gorenstein \( (\mathcal{X} \cap \mathcal{Y}, \mathcal{Y})\)-injective modules, and gives some properties and equivalent characterizations. Moreover, the stability of strongly Gorenstein \( (\mathcal{X} \cap \mathcal{Y}, \mathcal{Y})\)-injective modules is studied, and the relation between it and Gorenstein \( (\mathcal{X} \cap \mathcal{Y}, \mathcal{Y})\)-injective modules is discussed.The construct of Rota-Baxter algebra on the Sweedler 4-dimensional Hopf algebra.https://zbmath.org/1449.160672021-01-08T12:24:00+00:00"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian"Li, Xuan"https://zbmath.org/authors/?q=ai:li.xuan"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin|li.xin.7|li.xin.12|li.xin.2|li.xin.3|li.xin.9|li.xin.10|li.xin.6|li.xin.11|li.xin.1|li.xin.13|li.xin.4|li.xin.14|li.xin.5"Zheng, Huihui"https://zbmath.org/authors/?q=ai:zheng.huihui"Li, Linhan"https://zbmath.org/authors/?q=ai:li.linhan"Zhang, Liangyun"https://zbmath.org/authors/?q=ai:zhang.liangyunSummary: The nontrivial Rota-Baxter operators with weight \(-1\) are constructed from the Sweedler four-dimensional Hopf algebra and its subalgebras.Strongly \(g (x)\)-nil-clean rings.https://zbmath.org/1449.160232021-01-08T12:24:00+00:00"Chen, Yining"https://zbmath.org/authors/?q=ai:chen.yining"Qin, Long"https://zbmath.org/authors/?q=ai:qin.longSummary: The concepts of strongly \(g (x)\)-nil-clean rings are introduced. The relations of several classes of strongly \(g (x)\)-clean rings are discussed. Some equivalent characters of such rings and strongly nil-clean rings are given. Moreover, basic properties of such rings are investigated. Properties of several kinds of strongly \(g (x)\)-nil-clean rings are studied.A class of local nonlinear triple higher derivable maps on triangular algebras.https://zbmath.org/1449.160792021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Dai, Lei"https://zbmath.org/authors/?q=ai:dai.leiSummary: Let \(\mathcal{U}\) be a 2-torsion free triangular algebra, \(\Omega = \{x \in \mathcal{U}:{x^2} = 0\}\) and \(D = \{{d_n}\}_{n \in \mathbb{N}}\) be a sequence mapping from \(\mathcal{U}\) into itself (without assumption of additivity). By using the method of algebraic decomposition, we prove that if
\[{d_n} (xyz) = \sum\limits_{i+j+k=n}{d_i} (x){d_j} (y){d_k} (z)\] for all \({n \in \mathbb{N}}\), \(x,y,z \in \mathcal{U}\) with \(xyz \in \Omega\), then \(D\) is a higher derivation.Pseudo-unitary matrices and the \(*\)-structures of Radford algebra.https://zbmath.org/1449.150472021-01-08T12:24:00+00:00"Li, Shiyu"https://zbmath.org/authors/?q=ai:li.shiyu"Zhou, Hainan"https://zbmath.org/authors/?q=ai:zhou.hainan"Shen, Wenjie"https://zbmath.org/authors/?q=ai:shen.wenjie"Chen, Huixiang"https://zbmath.org/authors/?q=ai:chen.huixiangSummary: The 8-dimensional Radford algebra over the complex number field is a Hopf algebra whose \(*\)-structures are determined by complex \(2\times 2\)-matrices \(A\) satisfying \(\bar AA = I\). Such matrices are called pseudo-unitary matrices. The two \(*\)-structures determined by two pseudo-unitary matrices are equivalent if and only if the two pseudo-unitary matrices satisfy an equivalence relation \(\sim\). In this paper, the pseudo-unitary \(2\times 2\)-matrices are studied and classified with respect to the equivalence relation \(\sim\). It is shown that any pseudo-unitary \(2\times 2\)-matrix is equivalent to the identity matrix with respect to \(\sim\). Consequently, up to the equivalence of \(*\)-structures, the 8-dimensional Radford algebra has a unique Hopf \(*\)-algebra structure.The prime congruences and the radical of congruences on semirings.https://zbmath.org/1449.161012021-01-08T12:24:00+00:00"Wu, Ya'nan"https://zbmath.org/authors/?q=ai:wu.yanan"Ren, Miaomiao"https://zbmath.org/authors/?q=ai:ren.miaomiaoSummary: In this paper, we study congruences on commutative idempotent semirings. We give some results on prime congruences and the radical of congruences. Also, we reveal the relation between the radical of a congruence \(\rho\) and the set of the prime congruences containing \(\rho\).On primely polar rings.https://zbmath.org/1449.160242021-01-08T12:24:00+00:00"Li, Jiechen"https://zbmath.org/authors/?q=ai:li.jiechen"Chen, Huanyin"https://zbmath.org/authors/?q=ai:chen.huanyinSummary: A element \(a\) of a ring \(R\) is called primely polar if there exists \({p^2} = p \in \mathrm{comm}^2 (a)\) such that \(a+p \in U (R)\) and \(ap \in P (R)\). A ring \(R\) is said to be primely polar in case that every element of \(R\) is primely polar. This paper connects the primely polar rings with other related rings, proves that a commutative strongly \(\pi\)-regular ring is primely polar and a primely polar ring is strongly \(\pi\)-regular. Furthermore, it investigates the characteristics of primely polar rings in Drazin inverses. The results show that a ring \(R\) is primely polar if and only if for any \(a \in R\), there exists \({e^2} = e \in \mathrm{comm} (a)\) such that \(a-e \in U (R)\), \(ae \in P (R)\), if and only if for any \(a \in R\) there exists \(b \in \mathrm{comm} (a)\) such that \(b = {b^2}a\), \(a-{a^2}b \in P (R)\).Results on \({\mathcal{C}_n}\)-Flat modules.https://zbmath.org/1449.160092021-01-08T12:24:00+00:00"Wang, Xi"https://zbmath.org/authors/?q=ai:wang.xi"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Shen, Lei"https://zbmath.org/authors/?q=ai:shen.leiSummary: \(\mathcal{C}_n\)-flat modules are defined and studied in this paper. A right \(R\)-module \(M\) is said to be \(\mathcal{C}_n\)-flat provided that \({\mathrm{Tor}}_1^R (M,C) = 0\) for every \(n\)-cotorsion left \(R\)-module \(C\). It is proved that \(M\) is flat if and only if \(M\) is \(\mathcal{C}_n\)-flat and \({\mathrm{fd}}_RM \le 1\); and \(\mathcal{C}_nF\) is closed with pure submodules and pure quotient modules. Moreover, \(\mathcal{C}_1F\) and \(\mathcal{C}F\) are identical to the class of flat modules. In addition, if \(R\) is a domain and then every \(\mathcal{C}_2\)-flat module is also flat module. Finally, \(R\) has weak global dimension \(\le n\) if and only if the \(n\)th-syzygy of any right \(R\)-module is \(\mathcal{C}_n\)-flat, and \(R\) is von Neumann regular ring if and only if every right module is \(\mathcal{C}_n\)-flat module.The second nonlinear mixed Lie triple derivations on factor von Neumann algebras.https://zbmath.org/1449.160872021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: Let \(\mathcal{M}\) be a factor von Neumann algebra with dim \(\mathcal{M} > 1\), and \(L:\mathcal{M} \to \mathcal{M}\) be the second nonlinear mixed Lie triple derivation, i.e., \(L\) satisfies \[L ([[A,B],C]_*) = [[L (A), B],C]_* + [[A, L (B)], C]_* + [[A, B], L (C)]_*\] for all \(A, B, C \in \mathcal{M}\). Then \(L\) is an additive \(*\)-derivation.On Gorenstein \(FI\)-injective modules.https://zbmath.org/1449.160272021-01-08T12:24:00+00:00"Chen, Dong"https://zbmath.org/authors/?q=ai:chen.dong"Hu, Kui"https://zbmath.org/authors/?q=ai:hu.kuiSummary: The concept of Gorenstein \(FI\)-injective modules is introduced, which is a particular class between injective and Gorenstein injective modules. Some properties of Gorenstein \(FI\)-injective modules are discussed. A sufficient condition for Gorenstein \(FI\)-injective module to be injective is given, and semihereditary rings are characterized in terms of Gorenstein \(FI\)-injective modules. Finally, the Gorenstein \(FI\)-injective dimension is defined, and a sufficient condition for the existence of Gorenstein \(FI\)-injective (pre)envelope is given.\(n\)-strongly Gorenstein projective modules with respect to cotorsion pairs.https://zbmath.org/1449.160312021-01-08T12:24:00+00:00"Wang, Zhanping"https://zbmath.org/authors/?q=ai:wang.zhanping"Deng, Yaping"https://zbmath.org/authors/?q=ai:deng.yapingSummary: This paper introduces the \(n\)-strongly Gorenstein projective modules with respect to complete and hereditary cotorsion pairs, and discusses their homological properties. Some equivalent characterizations of \(n\)-strongly Gorenstein projective modules with respect to cotorsion pairs are given.Lazy 2-cocycle on Radford biproduct Hom-Hopf algebra.https://zbmath.org/1449.160622021-01-08T12:24:00+00:00"Ma, Tianshui"https://zbmath.org/authors/?q=ai:ma.tianshui"Zheng, Huihui"https://zbmath.org/authors/?q=ai:zheng.huihuiSummary: In this paper, we study lazy 2-cocycle on Radford's biproduct Hom-Hopf algebra. By using the twisting method, we mainly investigate the relations between the left Hom-2-cocycles \(\sigma\) on \( (B, \beta)\) and \(\overline \sigma\) on \( (B_ \times^\# H, \beta \otimes \alpha)\) which generalize the corresponding results in the case of usual Hopf algebras.Projective lattices of tiled orders.https://zbmath.org/1449.160362021-01-08T12:24:00+00:00"Zhuravlev, V. M."https://zbmath.org/authors/?q=ai:zhuravlev.viktor-mikhailovich"Tsyganivs'ka, I. M."https://zbmath.org/authors/?q=ai:tsyganivska.iryna-mSummary: Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, \textit{V. A. Jategaonkar} [Trans. Am. Math. Soc. 196, 313--330 (1974; Zbl 0292.16021)] proved that for every \(n \geq 2\), there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring \(\mathcal{O}\) of finite global dimension which lie in \(M_n(K)\), where \(K\) is a field of fractions of a commutatively discrete valuation ring \(\mathcal{O}\). \textit{H. Fujita} [Trans. Am. Math. Soc. 322, No. 1, 329--342 (1990; Zbl 0716.16009)] described the reduced tiled orders in Mn(D) of finite global dimension for n = 4, 5. \textit{V. Zhuravlev} and \textit{D. Zhuravlyov} [Algebra Discrete Math. 14, No. 2, 323--336 (2012; Zbl 1288.16010)] described reduced tiled orders in \(M_n(D)\) of finite global dimension for \(n =6\). This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let \(A\) be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form \(M_1f_1 + M_2f_2 + \cdots + M_sf_s\), where \(M_i\) is irreducible lattice, \(f_i\) is a vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.Strong \(k\)-commutativity preserving maps on prime rings with characteristic 2.https://zbmath.org/1449.160702021-01-08T12:24:00+00:00"Jia, Juan"https://zbmath.org/authors/?q=ai:jia.juan"Qi, Xiaofei"https://zbmath.org/authors/?q=ai:qi.xiaofeiSummary: Let \(\mathcal{R}\) be a unital prime ring of characteristic 2 containing a nontrivial idempotent. Assume that \(f:\mathcal{R} \to \mathcal{R}\) is a surjective map and \(k = 2,3\). Then \(f\) satisfies \([f (x), f (y)]_k = [x,y]_k = [[x,y]_{k-1}, y]\) for all \(x,y \in \mathcal{R}\) if and only if there exist a map \(\mu:\mathcal{R} \to \mathcal{C}\) and an element \(\lambda \in \mathcal{C}\) with \(\lambda^{k+1} = 1\) such that \(f (x) = \lambda x + \mu (x)\) for all \(x \in \mathcal{R}\), where \(\mathcal{C}\) is the extended centroid of \(\mathcal{R}\).Quasi-orthodox quasi completely regular semirings.https://zbmath.org/1449.160982021-01-08T12:24:00+00:00"Maity, S. K."https://zbmath.org/authors/?q=ai:maity.sunil-kumar"Ghosh, R."https://zbmath.org/authors/?q=ai:ghosh.rituparnaSummary: The aim of this paper is to characterize a class of additively quasi regular semirings which are subdirect products of an idempotent semiring and a \(b\)-lattice of quasi skew-rings.On uniformly strongly prime ternary rings.https://zbmath.org/1449.160442021-01-08T12:24:00+00:00"Salim, Md."https://zbmath.org/authors/?q=ai:salim.mohamed-ahmed-m"Dutta, T. K."https://zbmath.org/authors/?q=ai:dutta.tapan-kumarSummary: In this paper we introduce the notions of uniformly strongly prime ternary ring and uniformly strongly prime ideal and study them. We show that the class of all uniformly strongly prime ternary rings is a special class.On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras.https://zbmath.org/1449.160392021-01-08T12:24:00+00:00"Ashraf, Mohammad"https://zbmath.org/authors/?q=ai:ashraf.mohammad"Wani, Bilal Ahmad"https://zbmath.org/authors/?q=ai:wani.bilal-ahmadSummary: The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let \(R\) be a 2-torsion free semiprime *-ring. In this paper it is shown that, if \(R\) admits an additive mapping \(D : R \mapsto R\) satisfying either \(D(xyx) = D(xy)x^* + xyD(x)\) for all \(x,y\in R\), or \(D(xyx) = D(x)y^* x^* + xD(yx)\) for all pairs \(x,y \in R\), then \(D\) is a *-derivation. Moreover, this result makes it possible to prove that if \(R\) satisfies \(2D(x n) = D(x^{n-1})x^* + x^{n-1} D(x) + D(x)(x^*)^{n-1} + xD(x^{n-1})\) for all \(x\in R\) and some fixed integer \(n\geq 2\), then \(D\) is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras \(\mathcal{A(H)}\). In particular, we prove that if \(\mathcal{H}\) is a real or complex Hilbert space, with \(\dim(\mathcal{H}) > 1\), admitting a linear mapping \(D : \mathcal{A(H)}\mapsto\mathcal{B(H)}\) (where \(\mathcal{B(H)}\) stands for the bounded linear operators) such that \[2D(A^n) = D(A^{n-1})A^* + A^{n-1} D(A) + D(A)(A^*)^{n-1} + AD(A^{n-1})\] for all \(A \in\mathcal{A(H)}\), then \(D\) is Jordan *-derivation.On modules with insertion factor property.https://zbmath.org/1449.160042021-01-08T12:24:00+00:00"Nghiem, Nguyen Dang Hoa"https://zbmath.org/authors/?q=ai:nghiem.nguyen-dang-hoa"Van, Sanh Nguyen"https://zbmath.org/authors/?q=ai:van.sanh-nguyen"Bac, Nguyen Trong"https://zbmath.org/authors/?q=ai:bac.nguyen-trong"Somsup, Chitlada"https://zbmath.org/authors/?q=ai:somsup.chitladaSummary: A right ideal \(I\) of a ring \(R\) is an IFP right ideal if for any \(a, b \in R\), if \(ab \in I\), then \(aRb \subset I\). A ring \(R\) is called an IFP ring if 0 is an IFP ideal of \(R\). In this paper, we introduce the notion of IFP modules as a generalization of IFP rings. Many good properties of IFP rings can be transferred to IFP modules. We also give a generalization of Anderson's theorem.Nil subnear-ring of a non-singular Goldie near-ring and its nilpotency.https://zbmath.org/1449.160932021-01-08T12:24:00+00:00"Das, G. C."https://zbmath.org/authors/?q=ai:das.gireen-ch"Chowdhury, KC."https://zbmath.org/authors/?q=ai:chowdhury.kc"Das, P."https://zbmath.org/authors/?q=ai:das.pali|das.poulami|das.pitambar|das.pratibhamoy|das.priyanka|das.panchanan|das.prabin|das.pratap-kumar|das.purna-candra|das.prabir|das.pranab-k-ii|das.pradyut|das.pramode-k|das.prodip-kumar|das.pratyayananda|das.pinaki|das.ponkog-kumar|das.proloy|das.praloy|das.pankaja|das.priyadwip|das.paritosh-chandra|das.purabi|das.purnima|das.pankaj-kumar|das.prakash-kumar|das.prashant|das.partha-pratim|das.parthasakha|das.praggya|das.praachi|das.pulak|das.pranabesh|das.premadhis|das.partha-s|das.pratulananda|das.pradip-kumar|das.priyam|das.pritha|das.prasenjit|das.prosenjit|das.phonindra-nath|das.pintu|das.pritam|das.phullendu|das.pradeep|das.payel|das.prohelika|das.pankaz|das.p-sivaramakrishna|das.pravangsu-sekhar|das.purnendu|das.pankaj-k|das.pramod-kumar|das.provash|das.parthasarathi|das.prosanjit|das.paramita|das.priyanshu|das.prasanta-k|das.prasun|das.parimalSummary: We present here an important result that a nil subnear-ring of a non-singular strictly left Goldie near-ring is nilpotent. It is to be noted that the essentiality of left near-ring subgroup here, arises as crucial from its feeble nature. In contrast to such a result in ring theory, the crucial role played by substructures already mentioned appears here with very fascinating distinctiveness.Weak near-Armendariz rings.https://zbmath.org/1449.160942021-01-08T12:24:00+00:00"Pazoki, Maryam"https://zbmath.org/authors/?q=ai:pazoki.maryam"Najafian, Mehrab"https://zbmath.org/authors/?q=ai:najafian.mehrabSummary: In this paper, we introduce a concept that is called weak near-Armendariz ring, which is a generalization of both Armendariz rings and 2-primal rings. We show the basic properties of weak near-Armendariz rings and prepare some typical examples. It is proved that direct limit of a direct system of weak near-Armendariz rings is also a weak near-Armendariz ring.On the unit groups of commutative group algebras.https://zbmath.org/1449.160482021-01-08T12:24:00+00:00"Kuneva, Velika"https://zbmath.org/authors/?q=ai:kuneva.velika-n"Mollov, Todor"https://zbmath.org/authors/?q=ai:mollov.todor-zh"Nachev, Nako"https://zbmath.org/authors/?q=ai:nachev.nako-aSummary: Let \(RG\) be the group algebra of an abelian group \(G\) over a commutative indecomposable ring \(R\) with identity of prime characteristic p and \(U(RG)\) be the unit group of \(RG\). \textit{R. B. Warfield} jun. [Bull. Am. Math. Soc. 78, 88--92 (1972; Zbl 0231.13004)] introduced the concept \(KT\)-module \(M\) over a discrete valuation ring and invariants \(W_{\alpha,q}(M)\), denoted by \(h(\alpha,M)\), for an arbitrary limit ordinal \(\alpha\) and prime \(q\). \textit{P. V. Danchev} [J. Algebra Appl. 8, No. 6, 829--836 (2009; Zbl 1183.16031)] calculated the values \(W_{\alpha,q}(U(RG))\), when the quotient group \(G_t/G_p\) is finite (Proposition 10), where \(G_t\) is the torsion subgroup of \(G\) and \(G_p\) is the \(p\)-component of \(G\). ln the present paper we establish that Proposition 10 is not valid and does not have a sense, since for an arbitrary prime \(r\ne p\) and \(G=A\times B\), where \(A\) is a \(p\)-group and \(B\) is a cyclic group of order \(r^n\), \(n\in\mathbb{N}\), we obtain the contradiction that \(W_{\alpha,q}(U(RG))\) is a fraction when \(\zeta_{p^n}\) is a primitive \(p^n\)-th root of identity over \(R\) such that \(\zeta_{r^n}\) is not a root of a polynomial over \(R\) of degree les than \(r^n\).Graded Matlis cotorsion modules and graded Matlis domains.https://zbmath.org/1449.160902021-01-08T12:24:00+00:00"Wu, Xiaoying"https://zbmath.org/authors/?q=ai:wu.xiaoying"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Xie, Yajing"https://zbmath.org/authors/?q=ai:xie.yajingSummary: Let \(R\) be a \(G\)-graded integral domain. The notions of graded \(h\)-divisible \(R\)-module, graded Matlis cotorsion \(R\)-module and graded Matlis domain are introduced. It is shown in this paper that: (1) if \(M\) is a graded module, then \({\mathrm{gr}}-{\mathrm{pd}}_R (M) \le 1\) if and only if \({\mathrm{Ext}}_R^1 (M,D) = 0\) for each graded \(h\)-divisible module \(D\); (2) \(M\) is a graded Matlis cotorsion module if and only if \(M (\sigma)\) is a graded Matlis cotorsion module for each \(\sigma \in G\); (3) \(R\) is a graded Matlis domain if and only if the pair (\({\mathrm{gr}}-{P_1}\), \({\mathrm{gr}}-LC\)) forms a graded cotorsion theory, where \({\mathrm{gr}}-{P_1}\) is the class of graded modules of graded projective dimension at most one and \({\mathrm{gr}}-LC\) is the class of graded \(h\)-divisible modules.On \(\alpha \)-modules and their applications.https://zbmath.org/1449.160072021-01-08T12:24:00+00:00"Hasan, Ayazul"https://zbmath.org/authors/?q=ai:hasan.ayazul"Rafiquddin"https://zbmath.org/authors/?q=ai:rafiquddin.m"Ahmad, Mohammad Fareed"https://zbmath.org/authors/?q=ai:ahmad.mohammad-fareedSummary: A \(QTAG\)-module \(M\) is an \(\alpha \)-module, where \(\alpha \) is a limit ordinal, if \(M/{H_\beta}(M)\) is totally projective for every ordinal \(\beta < \alpha \). Here we show that totally projective modules and \(\alpha \)-modules of length \(\alpha \), where \(\alpha \) has cofinality \(\omega \), are projectively socle-regular. We also show that a summable \({\omega_1}\)-module needs not to be a direct sum of countably generated modules, where \({\omega_1}\) is the first uncountable ordinal.Fuzzy structure space of semirings and \(\Gamma \)-semirings.https://zbmath.org/1449.161022021-01-08T12:24:00+00:00"Goswami, Sarbani Mukherjee"https://zbmath.org/authors/?q=ai:goswami.sarbani-mukherjee"Mukhopadhyay, Arup"https://zbmath.org/authors/?q=ai:mukhopadhyay.arup-ranjan|mukhopadhyay.arup-kumar"Sardar, Sujit Kumar"https://zbmath.org/authors/?q=ai:sardar.sujit-kumarSummary: The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a \(\Gamma\)-semiring. We study separation axioms, compactness, etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a \(\Gamma\)-semiring \(S\) by using the nice interplay between \(S\) and its left operator semiring \(L\).New results on generalized \( (m,n)\)-Jordan derivations over semiprime rings.https://zbmath.org/1449.160412021-01-08T12:24:00+00:00"Ghosh, Arindam"https://zbmath.org/authors/?q=ai:ghosh.arindam"Prakash, Om"https://zbmath.org/authors/?q=ai:prakash.omSummary: Let \(m\) and \(n\) be positive integers such that \(m \ne n\), and \(R\) be an \(mn (m + n)|m - n|\)-torsion free semiprime ring. We prove that every generalized \( (m, n)\)-Jordan derivation over \(R\) is a derivation which maps \(R\) into its center.A note on connected cofiltered coalgebras, conilpotent coalgebras and Hopf algebras.https://zbmath.org/1449.160582021-01-08T12:24:00+00:00"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.liSummary: This note gives a proof that a connected coaugmented cofiltered coalgebra is a conilpotent coalgebra and thus a connected coaugmented cofiltered bialgebra is a Hopf algebra. This is applied in particular to a connected coaugmented cograded coalgebra and a connected coaugmented cograded bialgebra.The monad of \(\Omega\)-\(\mathrm{Cat}\) category.https://zbmath.org/1449.180012021-01-08T12:24:00+00:00"Zhao, Na"https://zbmath.org/authors/?q=ai:zhao.na"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jingSummary: In this paper, we construct a monad of \(\Omega\)-\(\mathrm{Cat}\) category, and prove that \(\Omega\)-\(\mathrm{Poset}\) category is isomorphic to \(\Omega\)-\(\mathrm{Cat}^T\) category.Globalizations for partial Hom-module algebras.https://zbmath.org/1449.160592021-01-08T12:24:00+00:00"Guo, Huaiwen"https://zbmath.org/authors/?q=ai:guo.huaiwen"Chen, Quanguo"https://zbmath.org/authors/?q=ai:chen.quanguoSummary: Using the associator deformation method, we consider the globalization problem for partial Hom-module algebras, and prove that every partial Hom-module algebra has a globalization.McCoy rings of quadratic truncated Nakayama algebras.https://zbmath.org/1449.160572021-01-08T12:24:00+00:00"Zhou, Yuye"https://zbmath.org/authors/?q=ai:zhou.yuye"Cheng, Zhi"https://zbmath.org/authors/?q=ai:cheng.zhiSummary: In this paper, we study some kind of matrix rings. We give a condition for a kind of matrix ring over ring \(R\) to be a McCoy ring. Moreover, we use this result to give a necessary and sufficient condition for a quadratic truncated Nakayama algebra to be a McCoy ring.The structure of Q-clean ring and its application.https://zbmath.org/1449.160732021-01-08T12:24:00+00:00"Sun, Xiaoqing"https://zbmath.org/authors/?q=ai:sun.xiaoqing"Liu, Xia"https://zbmath.org/authors/?q=ai:liu.xia"Hua, Xiujuan"https://zbmath.org/authors/?q=ai:hua.xiujuanSummary: The notions of Q-clean elements in a ring and Q-clean ring are defined. Some basic properties of Q-clean rings are obtained. In particular, it is shown that every element in a Q-clean ring with 2 invertible is a sum of a quasi-invertible element and a square root of 1. Let \(R\) be a ring in which every element \(x \in R, x = xux\) where \(u \in R_q^{-1}\). Then \(R\) is Q-clean. Furthermore, we prove that the ideals of a Q-clean ring are also Q-clean.Commutative zero point \(\xi \)-Lie higher derivable maps on triangular algebras.https://zbmath.org/1449.160812021-01-08T12:24:00+00:00"Fei, Xiuhai"https://zbmath.org/authors/?q=ai:fei.xiuhai"Zhang, Haifang"https://zbmath.org/authors/?q=ai:zhang.haifang"Lu, Cuixian"https://zbmath.org/authors/?q=ai:lu.cuixianSummary: Let \(\mathcal{U}\) be a triangular algebra over a number field \(\mathcal{F}\). If \(D = \{d_k\}_{k \in \mathbb{N}}\) is a commutative zero point \(\xi\)-Lie \( (\xi \ne 1)\) higher derivable mapping from \(\mathcal{U}\) into itself with \({d_k} (1) = 0\) \((\forall k \in \mathbb{N}^+)\), then \(D\) is a higher derivation.Construction of semisimple categories over weak Hopf algebras.https://zbmath.org/1449.160642021-01-08T12:24:00+00:00"Zhang, Xiaohui"https://zbmath.org/authors/?q=ai:zhang.xiaohui"Wu, Hui"https://zbmath.org/authors/?q=ai:wu.huiSummary: We study the properties of the category of the Yetter-Drinfeld modules over a weak Hopf algebra, and give a sufficient condition for the Yetter-Drinfeld category to be semisimple.A class of non-global derivable maps on prime rings.https://zbmath.org/1449.160842021-01-08T12:24:00+00:00"Kong, Liang"https://zbmath.org/authors/?q=ai:kong.liang"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhua|zhang.jianhua.1Summary: Let \(\mathscr{R}\) be a unital prime ring containing a nontrivial idempotent, \(\mathscr{Q} =\{T \in \mathscr{R}: {T^2} = 0\}\), and \(\mathscr{R} \to \mathscr{R}\) be a map (without additive assumption). Using the method of algebraic decomposition, we prove that if \(\delta (AB) = \delta (A)B + A\delta (B)\) for any \(A,B \in \mathscr{R}\) with \([A, B]B \in \mathscr{Q}\), then \(\delta\) is an additive derivation, where \([A, B] = AB - BA\) is the Lie product.E-lifting modules relative to fully invariant submodules.https://zbmath.org/1449.160012021-01-08T12:24:00+00:00"Alizadeh, F."https://zbmath.org/authors/?q=ai:alizadeh.fereydoon|alizadeh.farida-ch|alizadeh.farid"Hosseinpour, M."https://zbmath.org/authors/?q=ai:hosseinpour.mansoureh|hosseinpour.mehrab"Kamali, Z."https://zbmath.org/authors/?q=ai:kamali.zeinab|kamali.zahraSummary: In this paper, we introduce the notion FI-e-lifting modules which is a proper generalization of lifting (e-lifting) modules. Then we give some characterizations and properties of e-lifting and FI-e-lifting modules. We provide a decomposition of any e-lifting modules. It is shown that every finite direct sum of FI-e-lifting modules is FI-e-liftingSome studies on quotient semimodules over commutative semirings.https://zbmath.org/1449.160952021-01-08T12:24:00+00:00"Han, Shu"https://zbmath.org/authors/?q=ai:han.shu"Li, Yuying"https://zbmath.org/authors/?q=ai:li.yuyingSummary: On the basis of classical algebra, the congruence relations and the congruence classes over commutative semirings are discussed. On the basis of them and the cancelable semimodules, dense subsemimodules, order relations and their minimal elements, we prove some properties of quotient semimodules, and then define the invariant subsemimodules on quotient semimodules, on which we begin to discuss quotient transformations and study their properties using homomorphic isomorphisms.Annihilator condition on power values of commutators with derivations.https://zbmath.org/1449.160422021-01-08T12:24:00+00:00"Huang, Shuliang"https://zbmath.org/authors/?q=ai:huang.shuliangSummary: Let \(R\) be a prime ring with center \(Z (R)\), \(I\) be a nonzero ideal of \(R\), \(d\) be a nonzero derivation of \(R\) and \(0 \ne a \in R\). In the present paper, our object is to study the situation \(a[d ({x^k}), {x^k}]^n \in Z (R)\) for all \(x \in I\) under certain conditions, where \(n (\ge 1)\), \(k (\ge 1)\) are fixed integers.On generalized Lie derivations.https://zbmath.org/1449.160772021-01-08T12:24:00+00:00"Bennis, Driss"https://zbmath.org/authors/?q=ai:bennis.driss"Vishki, Hamid Reza Ebrahimi"https://zbmath.org/authors/?q=ai:vishki.hamid-reza-ebrahimi"Fahid, Brahim"https://zbmath.org/authors/?q=ai:fahid.brahim"Bahmani, Mohammad Ali"https://zbmath.org/authors/?q=ai:bahmani.mohammad-aliSummary: In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by \textit{D. Benkovič} [Linear Algebra Appl. 434, No. 6, 1532--1544 (2011; Zbl 1216.16032)]. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras.An intersection condition for graded prime submodules in \(gr\)-multiplication modules.https://zbmath.org/1449.130022021-01-08T12:24:00+00:00"Al-Zoubi, Khaldoun"https://zbmath.org/authors/?q=ai:al-zoubi.khaldoun"Qarqaz, Feda'a"https://zbmath.org/authors/?q=ai:qarqaz.fedaaIn this paper, the authors study graded modules over \(G\)-graded commutative rings, where \(G\) is a group. Especially, they study condition \((*)\) for a gr-multiplication module \(M\): every prime submodule of \(M\) containing an intersection of a family of gr-submodules of \(M\), contains one of the submodules in this family. The class of gr-multiplication modules satisfying condition \((*)\) contains the gr-Artinian modules, it is contained in the class of gr-simple modules, and it is closed under images of graded homomorphisms and under localizations. Among other results, the authors prove that a graded \(R\)-module M over a \(G\)-graded ring \(R\) is a a gr-multiplication \(R\)-module if and only if every graded submodule \(N\) of \(M\) is uniquely determined by the annihilator Ann\(_ R(M/N)\).
Reviewer: Moshe Roitman (Haifa)Super associative Yang-Baxter equation in associative superalgebras.https://zbmath.org/1449.160662021-01-08T12:24:00+00:00"An, Huihui"https://zbmath.org/authors/?q=ai:an.huihuiSummary: In this paper, we study the super associative Yang-Baxter equation in associative superalgebras. Firstly, we give the definition of Rota-Baxter operator and \(\mathcal{O}\)-operator in an associative superalgebra. We also give the relation between the odd Rota-Baxter operator in an associative superalgebra and Lie superalgebra. Then, we give the relation between the solution of super associative Yang-Baxter equation and the \(\mathcal{O}\)-operator in an associative superalgebra. At last, we give the relation between the solution of super associative Yang-Baxter equation and super 2-cocycle in an associative superalgebra.Green's D-relation on a semiring CR\( (n,1)\).https://zbmath.org/1449.160972021-01-08T12:24:00+00:00"Lian, Lifeng"https://zbmath.org/authors/?q=ai:lian.lifengSummary: This paper studied the Green's relation of semiring whose additive semigroup should be a band and multiplicative semigroup should be a completely regular semigroup. The characterization of relations of \({\mathrm{\dot D}} \bigcap \overset{+}{\mathrm{D}}\), \({\mathrm{\dot D}} \bigcap {\overset{+}{\mathrm{L}}}\), \({\mathrm{\dot D}} \bigcap \overset{+}{\mathrm{R}}\) is carried out and sufficient and necessary conditions for \({\overset{+}{\mathrm{D}}} \bigcap {\overset{+}{\mathrm{D}}}\) to be a congruence relation is given.Solution to the Clebsch-Gordan problem for two classes of self-injective algebras with finite representation type.https://zbmath.org/1449.160352021-01-08T12:24:00+00:00"Yang, Shiying"https://zbmath.org/authors/?q=ai:yang.shiying"Yu, Xiaolan"https://zbmath.org/authors/?q=ai:yu.xiaolanSummary: The finite dimensional representation categories of two classes of self-injective algebras with finite representation type can be equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. Then, the Clebsch-Gordan problem is solved.Cleanness of matrix subrings involving prime radicals.https://zbmath.org/1449.160562021-01-08T12:24:00+00:00"Li, Jiechen"https://zbmath.org/authors/?q=ai:li.jiechen"Chen, Huanyin"https://zbmath.org/authors/?q=ai:chen.huanyinSummary: A ring \(R\) is strongly P-clean if every element in \(R\) is the sum of an idempotent element and a strong nilpotent element. In this paper, the strong P-cleanness of certain \(3 \times 3\) matrix rings is investigated. A ring \(R\) is strongly 2-P-clean if every element in \(R\) is the sum of a tripotent and a strong nilpotent element. Elementary results on such rings are derived. It is proved that a ring \(R\) is strongly 2-P-clean if and only if for any \(a \in R, {a^3} - a \in P (R)\) if and only if for any \(a \in R\), there exist orthogonal idempotent elements \(e, f \in R\) such that \(a - e + f \in P (R)\). Furthermore, related results about strongly 2-P-clean matrix subrings are obtained as well.McCoy property over prime radicals.https://zbmath.org/1449.160382021-01-08T12:24:00+00:00"Shehata, Shaimaa Sh"https://zbmath.org/authors/?q=ai:shehata.shaimaa-shSummary: In this paper we first introduce the notion of McCoy over prime radicals (MPR ring) which is a generalization of Armendariz over prime radicals (APR ring) and McCoy rings. Then we investigate their properties. For a ring \(R\), we prove that the \(n\)-by-\(n\) upper triangular matrix ring \({T_n} (R)\) is MPR. The structure and some of extensions of MPR rings will be further studied and investigated.