Recent zbMATH articles in MSC 13Chttps://zbmath.org/atom/cc/13C2021-07-26T21:45:41.944397ZWerkzeugSome algebraic structures with apartness, a reviewhttps://zbmath.org/1463.030452021-07-26T21:45:41.944397Z"Romano, Daniel A."https://zbmath.org/authors/?q=ai:romano.daniel-abrahamSummary: The logical environment of this text is the Intuitionistic Logic -- a logic without the principle of `Tertium non datur' and the principled-philosophical orientation of the Bishop's constructive algebra. This orientation enables us to construct algebraic structures on the relational structures of the type \((X,=,\ne)\) as basic carriers where `\(\ne\)' is a diversity relation/apartness. In the last forty years, many algebraic structures with apartness have been analyzed. In this article, we will expose the recapitulation of some of them and show some basic characteristics of the selected algebraic structures such as semigroups (ordered semigroup under co-quasiorder, semillatice-ordered semigroups, inverse semigroups, implicative semigroups, \(\Gamma\)-semigroups), groups (free abelian groups, ordered group under co-order) rings (commutative rings, semillatice-ordered semirings, \(\Gamma\)-semirings, modules over commutative rings).On matrix and lattice ideals of digraphshttps://zbmath.org/1463.052192021-07-26T21:45:41.944397Z"Damadi, Hamid"https://zbmath.org/authors/?q=ai:damadi.hamid"Rahmati, Farhad"https://zbmath.org/authors/?q=ai:rahmati.farhadSummary: Let \(G\) be a simple, oriented connected graph with \(n\) vertices and \(m\) edges. Let \(I(\mathcal{B})\) be the binomial ideal associated to the incidence matrix \(\mathcal{B}\) of the graph \(G\). Assume that \(I_L\) is the lattice ideal associated to the rows of the matrix \(\mathcal{B}\). Also let \(\mathcal{B}_i\) be a submatrix of \(\mathcal{B}\) after removing the \(i\)-th row. We introduce a graph theoretical criterion for \(G\) which is a sufficient and necessary condition for \(I(\mathcal{B})=I(\mathcal{B}_i)\) and \(I(\mathcal{B}_i)=I_L\). After that we introduce another graph theoretical criterion for \(G\) which is a sufficient and necessary condition for \(I(\mathcal{B})=I_L\). It is shown that the heights of \(I(\mathcal{B})\) and \(I(\mathcal{B}_i)\) are equal to \(n-1\) and the dimensions of \(I(\mathcal{B})\) and \(I(\mathcal{B}_i)\) are equal to \(m-n+1\); then \(I(\mathcal{B}_i)\) is a complete intersection ideal.The annihilating-submodule graph of modules over commutative ringshttps://zbmath.org/1463.052452021-07-26T21:45:41.944397Z"Ansari-Toroghy, Habibollah"https://zbmath.org/authors/?q=ai:ansari-toroghy.habibollah"Habibi, Shokoufeh"https://zbmath.org/authors/?q=ai:habibi.shokoufehSummary: Let \(M\) be a module over a commutative ring \(R\). In this paper, we continue our study of annihilating-submodule graph \(AG(M)\) which was introduced in \textit{H. Ansari-Toroghy} and \textit{S. Habibi} [Commun. Algebra 42, No. 8, 3283--3296 (2014; Zbl 1295.13016)]. \(AG(M)\) is a (undirected) graph in which a nonzero submodule \(N\) of \(M\) is a vertex if and only if there exists a nonzero proper submodule \(K\) of \(M\) such that \(NK = (0)\), where \(NK\), the product of \(N\) and \(K\), is defined by \((N : M)(K : M)M\) and two distinct vertices \(N\) and \(K\) are adjacent if and only if \(NK = (0)\). We obtain useful characterizations for those modules \(M\) for which either \(AG(M)\) is a complete (or star) graph or every vertex of \(AG(M)\) is a prime submodule of \(M\). Moreover, we study coloring of annihilating-submodule graphs.The strongly annihilating-submodule graph of a modulehttps://zbmath.org/1463.052522021-07-26T21:45:41.944397Z"Beyranvand, Reza"https://zbmath.org/authors/?q=ai:beyranvand.reza"Farzi-Safarabadi, Ahadollah"https://zbmath.org/authors/?q=ai:farzi-safarabadi.ahadollahSummary: In this paper, we define the notion of strongly annihilating-submodule graph of modules. This graph is a straightforward common generalization of the annihilating-submodule graph and the annihilating-ideal graph. In addition to providing the properties of this graph in general, we investigate the behavior of the graph when modules are reduced or divisible.Relative determinant of a bilinear modulehttps://zbmath.org/1463.111102021-07-26T21:45:41.944397Z"Koprowski, Przemysław"https://zbmath.org/authors/?q=ai:koprowski.przemyslawSummary: The aim of the paper is to generalize the (ultra-classical) notion of the determinant of a bilinear form to the class of bilinear forms on projective modules without assuming that the determinant bundle of the module is free. Successively it is proved that this new definition preserves the basic properties, one expects from the determinant. As an example application, it is shown that the introduced tools can be used to significantly simplify the proof of a recent result by B. Rothkegel.On strongly prime submoduleshttps://zbmath.org/1463.130052021-07-26T21:45:41.944397Z"Azizi, Abdulrasool"https://zbmath.org/authors/?q=ai:azizi.abdulrasoolSummary: Let \(R\) be a commutative ring with identity and \(M\) an \(R\)-module. A proper submodule \(N\) of \(M\) is called strongly prime (resp. strongly semiprime), if \(((N+Rx):M)y\subseteq N\) (resp. \(((N+Rx):M)x\subseteq N\)) for \(x,y\in M\) implies that \(x\in N\) or \(y\in N\) (resp. \(x\in N\)). Strongly prime and strongly semiprime submodules are studied, in this paper.On almost fuzzy prime ideals and sub-moduleshttps://zbmath.org/1463.130062021-07-26T21:45:41.944397Z"Bataineh, Malik"https://zbmath.org/authors/?q=ai:bataineh.malik"Al-Areqie, Akram"https://zbmath.org/authors/?q=ai:al-areqie.akramSummary: The purpose of this paper is to present some characterizations of almost fuzzy prime ideals and sub modules; and to introduce certain innovative fuzzy prime ideals and sub modules respectively. Moreover, every fuzzy ideal in R is an almost prime ideal for a quasi-local ring \( (R, M) \) with \( M^{2} = 0 \). Moreover, prime sub module \(\mu_N\) in a finitely generated multiplication \(R\)-module M and fuzzy prime ideal \(I_\mu\) in R have been constructed.A short note on prime submoduleshttps://zbmath.org/1463.130152021-07-26T21:45:41.944397Z"Azami, Jafar"https://zbmath.org/authors/?q=ai:azami.jafarSummary: Let \(R\) be a commutative ring with identity and \(M\) be a unital \(R\)-module. A proper submodule \(N\) of \(M\) with \(N:_RM=\mathfrak p\) is said to be prime or \(\mathfrak p\)-prime \((\mathfrak p\) a prime ideal of \(R)\) if \(rx\in N\) for \(r\in R\) and \(x\in M\) implies that either \(x\in N\) or \(r\in\mathfrak p\). In this paper we study a new equivalent conditions for a minimal prime submodules of an \(R\)-module to be a finite set, whenever \(R\) is a Noetherian ring. Also we introduce the concept of arithmetic rank of a submodule of a Noetherian module and we give an upper bound for it.CF-modules over commutative rings.https://zbmath.org/1463.130162021-07-26T21:45:41.944397Z"Najim, Ahmed"https://zbmath.org/authors/?q=ai:najim.ahmed"Charkani, Mohammed Elhassani"https://zbmath.org/authors/?q=ai:charkani.mohamed-elhassaniLet \(R\) be a commutative ring with unit and \(M\) an \(R\)-module. When \(M\simeq R/I_1\oplus\ldots\oplus R/I_n\) with \(I_1\subseteq I_2\subseteq\ldots\subseteq I_n\not=R\), where \(I_1,\ldots,I_n\) are ideals of \(R\), then \(I_1,\ldots,I_n\) are uniquely determined. In this case, we say that \(M\) is a CF-module of type \((I_1,\ldots,I_n)\). In general, the sum of two CF-modules is not necessarily a CF-module even in the case where \(R\) is local. The authors of this paper give some criterions in order that the sum of two CF-modules is a CF-module. They characterize CF-modules whose tensor product is a CF-module over a local ring.Some extensions of decomposition theorems in abelian groups. IIIhttps://zbmath.org/1463.130172021-07-26T21:45:41.944397Z"Qin, Xin"https://zbmath.org/authors/?q=ai:qin.xin"Liu, Heguo"https://zbmath.org/authors/?q=ai:liu.heguoSummary: On the basis of the Prüfer-Baer theorem of the bounded module over a principal ideal domain, this paper studies several basic problems about the algebraic linear transformation of some vector space (infinite dimensional). Let \(V\) be a vector space (infinite dimensional) over a field \(F\), \(\mathcal{A}\) be an algebraic linear transformation of \(V\):
\begin{itemize}
\item[(1)] Suppose that any linear transformation commuting with \(\mathcal{A}\) commutes also with a linear transformation \(\mathcal{B}\), then \(\mathcal{B} = f(\mathcal{A} )\), where \(f\) is a polynomial over \(F\).
\item[(2)] There exists a basis for \(V\) such that the matrix of \(\mathcal{A}\) relative to this basis has the rational canonical form (classical canonical form). Moreover the classical canonical form becomes the Jordan canonical form when \(F\) is algebraically closed.
\item[(3)] There exists the Jordan-Chevalley decomposition of \(\mathcal{A}\) when \(F\) is algebraically closed.
\end{itemize}
This result prevails for the perfect field in general. These results extend some theorems of finite dimensional vector spaces to infinite dimensional vector spaces.Some extensions of decomposition theorems in abelian groups. II.https://zbmath.org/1463.130182021-07-26T21:45:41.944397Z"Qin, Xin"https://zbmath.org/authors/?q=ai:qin.xin"Liu, Heguo"https://zbmath.org/authors/?q=ai:liu.heguo"Luo, Xiaoliang"https://zbmath.org/authors/?q=ai:luo.xiaoliangSummary: We first discuss two special quasi-cyclic modules over principal ideal domains and then investigate the structures of vector spaces of finite \(\mathbb{A}\)-width. We show that the poset of \(\mathbb{A}\)-invariant subspaces of a vector space of finite \(\mathbb{A}\)-width must satisfy the minimal condition, and give a sufficient and necessary condition for a vector space (as a \(F[\lambda]\)-module) to be a quasi-cyclic module.\par For part I see [the second author et al., ibid. 60, No. 6, 1065--1074 (2017; Zbl 1413.13009)].\(f\)-injective modules with respect to semidualizing moduleshttps://zbmath.org/1463.130192021-07-26T21:45:41.944397Z"Lan, Kaiyang"https://zbmath.org/authors/?q=ai:lan.kaiyang"Lu, Bo"https://zbmath.org/authors/?q=ai:lu.boSummary: Let \(R\) be a commutative ring and \(C\) a semidualizing \(R\)-module. The \(f\)-injective modules with respect to a semidualizing \(R\) module \(C\) is defined and studied. It is proved that a homomorphism \(F \to M\) of \(R\)-modules is an injective (\(f\)-injective) precover of \(M\) if and only if \({\mathrm{Hom}}_R(C, F) \to {\mathrm{Hom}}_R(C, M)\) is a \(C\)-injective (\(C\)-\(f\)-injective) precover.GI-modules and coreflexive complexeshttps://zbmath.org/1463.130202021-07-26T21:45:41.944397Z"Liu, Yanping"https://zbmath.org/authors/?q=ai:liu.yanpingSummary: A class of GI-modules, coreflexive complexes and their properties are investigated. It is proved that an artinian module has finite GI-dimension if and only if it is coreflexive as a complex. The GI-dimension of complexes is also studied and it is found that a complex homologically degreewise artinian and bounded to the left has finite GI-dimension if and only if it is coreflexive.The secondary radicals of submoduleshttps://zbmath.org/1463.130212021-07-26T21:45:41.944397Z"Ansari-Toroghy, Habibollah"https://zbmath.org/authors/?q=ai:ansari-toroghy.habibollah"Farshadifar, Faranak"https://zbmath.org/authors/?q=ai:farshadifar.faranak"Mahboobi-Abkenar, Farideh"https://zbmath.org/authors/?q=ai:mahboobi-abkenar.faridehSummary: Let \(R\) be a commutative ring with identity and let \(M\) be an \(R\)-module. In this paper, we will introduce the secondary radical of a submodule \(N\) of \(M\) as the sum of all secondary submodules of \(M\) contained in \(N\), denoted by \(sec^*(N)\), and explore the related properties. We will show that this class of modules contains the family of second radicals properly and can be regarded as a dual of primary radicals of submodules of \(M\).On $(k, n)$-closed submoduleshttps://zbmath.org/1463.130222021-07-26T21:45:41.944397Z"Celikel, Ece Yetkin"https://zbmath.org/authors/?q=ai:celikel.ece-yetkinSummary: Let $R$ be a commutative ring with $1\neq 0$ and $M$ an $R$-module. We will call a proper sub-module $N$ of $M$ as a semi $n$-absorbing submodule of $M$ if whenever $r\in R,m\in M$ with $r^nm\in N$, then $r^n\in (N:_RM)$ or $r^{n-1}m\in N$. We will say $N$ to be a $(k,n)$-closed submodule of $M$ if whenever $r\in R,m\in M$ with $r^km\in N$, then $r^n\in(N:_RM)$ or $r^{n-1}m\in N$. In this paper we introduce semi $n$-absorbing and $(k,n)$-closed submodules of modules over commutative rings, and investigate their basic properties.2-absorbing \(I\)-prime and 2-absorbing \(I\)-second submoduleshttps://zbmath.org/1463.130232021-07-26T21:45:41.944397Z"Farshadifar, Faranak"https://zbmath.org/authors/?q=ai:farshadifar.faranakSummary: Let \(R\) be a commutative ring and let \(I\) be an ideal of \(R\). In this paper, we will introduce the notions of 2-absorbing \(I\)-prime and 2-absorbing \(I\)-second submodules of an \(R\)-module \(M\) as a generalization of 2-absorbing and strongly 2-absorbing second submodules of \(M\) and explore some basic properties of these classes of modules.Developed Zariski topology-graphhttps://zbmath.org/1463.130242021-07-26T21:45:41.944397Z"Hassanzadeh-Lelekaami, Dawood"https://zbmath.org/authors/?q=ai:lelekaami.dawood-hassanzadeh"Karimi, Maryam"https://zbmath.org/authors/?q=ai:karimi.maryamSummary: In this paper, we introduce the developed Zariski topology-graph associated to an \(R\)-module \(M\) with respect to a subset \(X\) of the set of all quasi-prime submodules of \(M\) and investigate the relationship between the algebraic properties of \(M\) and the properties of its associated developed Zariski topology-graph.Some remarks on generalizations of classical prime submoduleshttps://zbmath.org/1463.130252021-07-26T21:45:41.944397Z"Zolfaghari, Masoud"https://zbmath.org/authors/?q=ai:zolfaghari.masoud"Koopaei, Mohammad Hossein Moslemi"https://zbmath.org/authors/?q=ai:koopaei.mohammad-hossein-moslemiSummary: Let \(R\) be a commutative ring with identity and \(M\) be a unitary \(R\)-module. Suppose that \(\varphi:S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace\) be a function where \(S(M)\) is the set of all submodules of \(M\). A proper submodule \(N\) of \(M\) is called an \((n-1,n)\text{-}\varphi\)-classical prime submodule, if whenever \(r_1,\ldots,r_{n-1}\in R\) and \(m\in M\) with \(r_1\ldots r_{n-1}m\in N\setminus\varphi(N)\), then \(r_1\ldots r_{i-1}r_{i+1}\ldots r_{n-1}m\in N\), for some \(i\in\lbrace 1,\ldots, n-1\rbrace(n\geqslant 3)\). In this work, \((n-1,n)\text{-}\varphi\)-classical prime submodules are studied and some results are established.On Nagata's result about height one maximal ideals and depth one minimal prime ideals. Ihttps://zbmath.org/1463.130262021-07-26T21:45:41.944397Z"Kemp, Paula"https://zbmath.org/authors/?q=ai:kemp.paula-a"Ratliff, Louis J. jun."https://zbmath.org/authors/?q=ai:ratliff.louis-j-jun"Shah, Kishor"https://zbmath.org/authors/?q=ai:shah.kishorSummary: It is shown that, for all local rings \((R, M)\), there is a canonical bijection between the set \(DO(R)\) of depth one minimal prime ideals \(\omega\) in the completion \(\widehat{R}\) of \(R\) and the set \(HO(R/Z)\) of height one maximal ideals \(\overline{M}'\) in the integral closure \((R/Z)'\) of \(R/Z\), where \(Z := Rad(R)\). Moreover, for the finite sets \(\mathbf{D} := \{V^* / V^* := (\widehat{R}/\omega)', \omega \in DO(R)\text{ and } H := \{V/V := (R/Z)'_{\overline{M}'}, \overline{M}' \in HO(R/Z)\}\):
\begin{itemize}
\item[(a)] The elements in \textbf{D} and \textbf{H} are discrete Noetherian valuation rings.
\item[(b)] \(\mathbf{D} = \{\widehat{V} \in \mathbf{H}\}\).
\end{itemize}On Nagata's result about height one maximal ideals and depth one minimal prime ideals. IIhttps://zbmath.org/1463.130272021-07-26T21:45:41.944397Z"Kemp, Paula"https://zbmath.org/authors/?q=ai:kemp.paula-a"Ratliff, Louis J. jun."https://zbmath.org/authors/?q=ai:ratliff.louis-j-jun"Shah, Kishor"https://zbmath.org/authors/?q=ai:shah.kishorSummary: We expand the theory of height one maximal ideals and depth one minimal prime ideals initiated by M. Nagata and continued by the authors in part I. A local ring is doho in case its completion has at least one depth one minimal prime ideal. We establish several families of doho local rings, prove that certain local rings associated with Rees valuation rings are doho, and complement a famous construction of Nagata by proving that each doho local domain \((R, M)\) of altitude \(\alpha \geq 2\) has a quadratic integral extension over-domain with precisely two maximal ideals, one of height \(\alpha\) and the other of height one.
For part I, see [the authors, ibid. 85, No. 3--4, 356--376 (2018; Zbl 1463.13026)].Uniqueness of primary decompositions in Laskerian le-moduleshttps://zbmath.org/1463.130282021-07-26T21:45:41.944397Z"Bhuniya, A. K."https://zbmath.org/authors/?q=ai:bhuniya.arun-k|bhuniya.anjan-kumar"Kumbhakar, M."https://zbmath.org/authors/?q=ai:kumbhakar.manotosh|kumbhakar.manasAn le-module $M$ over a commutative ring $R$ is a complete lattice ordered monoid $(M,+,\leqslant, e)$ with greatest element $e$ and module like action of $R$ on it. Such an $R$-module is called Laskerian if each submodule is a finite intersection of primary submodules. Theorems on primary decompositions of element of such modules are proved.A note on artinian primes and second moduleshttps://zbmath.org/1463.130292021-07-26T21:45:41.944397Z"Khaksari, Ahmad"https://zbmath.org/authors/?q=ai:khaksari.ahmadSummary: Prime submodules and artinian prime modules are characterized. Furthermore, some previous results on prime modules and second modules are generalized.Uniformly classical quasi-primary submoduleshttps://zbmath.org/1463.130302021-07-26T21:45:41.944397Z"Naderi, Mohammad Hassan"https://zbmath.org/authors/?q=ai:naderi.mohammad-hassanSummary: In this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. Several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. Then we investigate for a ring \(R\), when any finite intersection of (uniformly) primary submodules of any \(R\)-module is a (uniformly) classical quasi-primary submodule. Furthermore, the behavior of classical quasi-primary and uniformly classical quasi-primary submodules under localizations are studied. Also, we investigate the existence of (minimal) primary submodules containing classical quasi-primary submodules.The existence totally reflexive covershttps://zbmath.org/1463.130312021-07-26T21:45:41.944397Z"Heidarian, Zahra"https://zbmath.org/authors/?q=ai:heidarian.zahraSummary: Let \(R\) be a commutative Noetherian ring. We prove that over a local ring \(R\) every finitely generated \(R\)-module \(M\) of finite Gorenstein projective dimension has a Gorenstein projective cover \(\varphi:C\rightarrow M\) such that \(C\) is finitely generated and the projective dimension of \(\mathrm{Ker}\varphi\) is finite and \(\varphi\) is surjective.Local cohomology modules and relative Cohen-Macaulaynesshttps://zbmath.org/1463.130452021-07-26T21:45:41.944397Z"Zohouri, M. Mast"https://zbmath.org/authors/?q=ai:zohouri.m-mastSummary: Let \((R,\mathfrak{m})\) denote a commutative Noetherian local ring and let \(M\) be a finite \(R\)-module. In this paper, we study relative Cohen-Macaulay rings with respect to a proper ideal \(\mathfrak{a}\) of \(R\) and give some results on such rings in relation with Artinianness, Non-Artinianness of local cohomology modules and Lyubeznik numbers. We also present some related examples to this issue.Some characterizations of Gorenstein Prüfer domainshttps://zbmath.org/1463.130472021-07-26T21:45:41.944397Z"Xiong, Tao"https://zbmath.org/authors/?q=ai:xiong.tao.1Summary: It is well known that a domain \(R\) is a Prüfer domain if and only if every divisible module is FP-injective; if and only if every \(h\)-divisible module is FP-injective. In this paper, we introduce the concept of Gorenstein FP-injective modules, and show that a domain \(R\) is a Gorenstein Prüfer domain if and only if every divisible module is Gorenstein FP-injective; if and only if every \(h\)-divisible module is Gorenstein FP-injective.