Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132021-01-08T12:24:00+00:00WerkzeugGraded projective covers and commutative graded perfect rings.https://zbmath.org/1449.130052021-01-08T12:24:00+00:00"Xie, Yajing"https://zbmath.org/authors/?q=ai:xie.yajing"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Wu, Xiaoying"https://zbmath.org/authors/?q=ai:wu.xiaoyingSummary: Let \(G\) be a commutative group and let \(R =\mathop\bigoplus\limits_{\sigma \in G} {R_\sigma}\) be a commutative graded ring. In this paper, the equivalent characterizations about graded semiperfect rings and graded perfect rings are given. It is shown that: (1) Every finitely generated graded module over a graded local ring has a graded projective cover. (2) \(R\) is graded semiperfect if and only if \(R\) is a direct product of finite graded local rings. (3) \(R\) is graded perfect if and only if \(R/{J^g} (R)\) is graded semisimple and every nonzero graded module has a maximal graded submodule; if and only if every graded module satisfies the descending chain condition on cyclic submodules; if and only if \(R\) is a direct product of graded local rings \({R_i}\), and \({J^g} (R_i)\) is a \(T\)-nilpotent ideal. (4) If \(R\) is a strongly graded ring, then \(R\) is a graded perfect ring if and only if \({R_e}\) is a perfect ring.On the strong persistence property for monomial ideals.https://zbmath.org/1449.130122021-01-08T12:24:00+00:00"Reyes, Enrique"https://zbmath.org/authors/?q=ai:reyes.enrique|reyes.enrique-g"Toledo, Jonathan"https://zbmath.org/authors/?q=ai:toledo.jonathanSummary: Let \(I\) be the edge ideal associated to a graph with loops, a weighted graph or a clutter. In this paper we study when \(I\) has the strong persistence property, this is \((I^{k+1}:I) = I^k\) for each \(k \ge 1\).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.On nonnil-injective modules.https://zbmath.org/1449.130102021-01-08T12:24:00+00:00"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei.2|zhao.wei.6|zhao.wei.5|zhao.wei.4|zhao.wei|zhao.wei.1|zhao.wei.3"Zhang, Xiaolei"https://zbmath.org/authors/?q=ai:zhang.xiaoleiSummary: A commutative ring with a divided prime nil radical is called a \(\phi\)-ring. In this paper, we introduce \(\phi\)-torsion modules over a \(\phi\)-ring, investigate nonnil-injective modules and the nonnil-injective hull over a \(\phi\)-ring and characterize the nonnil-semisimple rings.\(n\)-absorbing submodules and multiplication modules over valuation rings.https://zbmath.org/1449.130072021-01-08T12:24:00+00:00"Pola, E. N."https://zbmath.org/authors/?q=ai:pola.e-n"Kianpi, M."https://zbmath.org/authors/?q=ai:kianpi.maurice"Diekouam, L. E. F."https://zbmath.org/authors/?q=ai:diekouam.l-e-fSummary: Let \(R\) be a commutative ring with \(1 \ne 0\), \(I\) a proper ideal of \(R\), \(M\) a unital \(R\)-module and \(N\) a proper submodule of \(M\). In this paper, we show that for any ideal \(I\) of a valuation ring \(R\), that is, a ring \(R\) such that for all \(a, b \in R\), \(a\) divides \(b\) or \(b\) divides \(a\), \(I\) is an \(n\)-absorbing ideal of \(R\) if and only if \(I[X]\) is an \(n\)-absorbing ideal of \(R[X]\). Moreover, we show that for an arbitrary \( (N:{_RM})\) is a strongly \(n\)-absorbing ideal of \(R\) whenever \(N\) is a strongly \(n\)-absorbing submodule of the \(R\)-module \(M\). Furthermore, we show that when \(R\) is a valuation ring, and \(N\) is an \(n\)-absorbing submodule of \(M\) then \( (N:{_RM})\) is an \(n\)-absorbing ideal of \(R\). Also we show that every proper submodule \(N\) of an \(R\)-module \(M\) is an \(n\)-absorbing submodule if an only if it is a strongly \(n\)-absorbing submodule. We give a necessary and sufficient condition under which a valuation domain \(R\) has Krull dimension \(\le 1\) as well as a necessary and sufficient condition under which for a non-zero ideal \(I\) of a valuation ring, one has \({\omega_R} (I) = n\) when \(n > 1\). We investigate multiplication modules. To be precise, for a multiplication module \(M\) over an arbitrary ring \(R\), we show that for every nonnegative integer \(n\), a proper submodule \(N\) is strongly \(n\)-absorbing if and only if \( (N:{_RM})\) is a strongly \(n\)-absorbing ideal in \(R\) and that for an \(n\)-absorbing submodule \(N\) a multiplication module \(M\) over a valuation ring \(R\), it holds that \( (M-rad (N))^n \subseteq N\). For any ring \(R\), this inclusion holds for \(n = 2\). We also show that \({\omega_M} (N) = {\omega_R} (N:{_RM})\) when \(R\) is a valuation ring.On graded 2-absorbing quasi primary ideals.https://zbmath.org/1449.130042021-01-08T12:24:00+00:00"Uregen, Rabia Nagehan"https://zbmath.org/authors/?q=ai:uregen.rabia-nagehan"Tekir, Unsal"https://zbmath.org/authors/?q=ai:tekir.unsal"Shum, Kar Ping"https://zbmath.org/authors/?q=ai:shum.kar-ping"Koc, Suat"https://zbmath.org/authors/?q=ai:koc.suatSummary: In this article, we introduce the concept of graded 2-absorbing quasi primary ideal which is a generalization of graded prime ideal. In the first part of the paper, we give many characterizations of these classes of ideals. In the second part, we study idealization of graded modules. In particular, we investigate graded radical in the idealization of a graded module. Furthermore, we determine \(h\)-dimension of idealization of a graded module and study some classical graded ideals such as graded maximal, graded prime, graded primary, graded 2-absorbing and graded 2-absorbing quasi primary ideals.Rings in which every regular locally principal ideal is projective.https://zbmath.org/1449.130132021-01-08T12:24:00+00:00"El Khalfaoui, Rachida"https://zbmath.org/authors/?q=ai:el-khalfaoui.rachida"Mahdou, Najib"https://zbmath.org/authors/?q=ai:mahdou.najibSummary: In this article, we study the class of rings in which every regular locally principal ideal is projective called \(LPP\)-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.A new characterization of P\(v\)MDS by Kronecker function rings.https://zbmath.org/1449.130112021-01-08T12:24:00+00:00"Zhou, Dechuan"https://zbmath.org/authors/?q=ai:zhou.dechuan"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Hu, Kui"https://zbmath.org/authors/?q=ai:hu.kuiSummary: In this paper, we show that if \(R\) is integrally closed, then \(R\) is a P\(v\)MD if and only if \(Kr (R, {v_c})\) is a \(w (R[X])\)-flat \(R[X]\)-module; if and only if \(Kr (R, {v_c})\) is a \(w\)-flat \(R\)-module. This is a generalization of that if \(R\) is integrally closed, then \(R\) is a Prüfer domain if and only if \(Kr (R, b)\) is a flat \(R[X]\)-module; if and only if \(Kr (R, b)\) is a flat \(R\)-module.Zero-divisor graphs of finite commutative rings: a survey.https://zbmath.org/1449.051372021-01-08T12:24:00+00:00"Singh, Pradeep"https://zbmath.org/authors/?q=ai:singh.pradeep-kumar"Bhat, Vijay Kumar"https://zbmath.org/authors/?q=ai:bhat.vijay-kumarSummary: This article gives a comprehensive survey on zero-divisor graphs of finite commutative rings. We investigate the results on structural properties of these graphs.A generalization of power graphs of commutative rings.https://zbmath.org/1449.051322021-01-08T12:24:00+00:00"Fatehi, M."https://zbmath.org/authors/?q=ai:fatehi.mahsa"Khashyarmanesh, K."https://zbmath.org/authors/?q=ai:khashyarmanesh.kazem"Afkhami, M."https://zbmath.org/authors/?q=ai:afkhami.mojganSummary: Let \(R\) be a commutative ring with non-zero identity. For a natural number \(n\), we associate a directed graph, denoted by \(\mathcal{P}_R^n\), with \({R^n}\backslash \{0\}\) as the vertex set and, for two distinct vertices \(X = ({x_1}, {x_2}, \dots, {x_n})\) and \(Y = ({y_1}, {y_2}, \dots, {y_n})\), we have an arc \(X \to Y\) in \(\mathcal{P}_R^n\) if and only if there exists an \(n \times n\) lower triangular matrix \(A = (a_{i_j})\) over \(R\) such that \(A{X^T} = {Y^T}\) and \({\mathrm{det}} (A) \in [{x_1}, {x_2}, \dots, {x_n}]\), where \([{x_1}, {x_2}, \dots, {x_n}]\) is a subsemigroup generated by the elements \({x_1}, {x_2}, \dots, {x_n}\) in the multiplicative semigroup \(R\). When we consider the ring \(R\) as a semigroup with respect to multiplication, the graph \(\mathcal{P}_R^1\) is the usual directed power graph (over a semigroup). Hence \(\mathcal{P}_R^n\) is a generalization of power graphs. In this paper we study some basic properties of \(\mathcal{P}_R^n\). Also we study the planarity, outer planarity and ring graph of \(\mathcal{P}_R^n\) and we determine the cases for which the graph \(\mathcal{P}_R^n\) has thickness 2.Stability and numerical approximation for a spacial class of semilinear parabolic equations on the Lipschitz bounded regions: Sivashinsky equation.https://zbmath.org/1449.130162021-01-08T12:24:00+00:00"Mesrizadeh, Mehdi"https://zbmath.org/authors/?q=ai:mesrizadeh.mehdi"Shanazari, Kamal"https://zbmath.org/authors/?q=ai:shanazari.kamalSummary: This paper aims to investigate the stability and numerical approximation of the Sivashinsky equations. We apply the Galerkin meshfree method based on the radial basis functions (RBFs) to discretize the spatial variables and use a group presenting scheme for the time discretization. Because the RBFs do not generally vanish on the boundary, they can not directly approximate a Dirichlet boundary problem by Galerkin method. To avoid this difficulty, an auxiliary parametrized technique is used to convert a Dirichlet boundary condition to a Robin one. In addition, we extend a stability theorem on the higher order elliptic equations such as the biharmonic equation by the eigenfunction expansion. Some experimental results will be presented to show the performance of the proposed method.A probabilistic approach toward finite commutative rings.https://zbmath.org/1449.130172021-01-08T12:24:00+00:00"Rehman, Shafiqur"https://zbmath.org/authors/?q=ai:rehman.shafiqur"Baig, Abdul Qudair"https://zbmath.org/authors/?q=ai:baig.abdul-qudair"Haider, Kamran"https://zbmath.org/authors/?q=ai:haider.kamranSummary: Let \(m\), \(n\) be positive integers and \(m \le n\). Suppose that we select two elements at random (with replacement) from the ring \({\mathbb{Z}_n}\). Then a question arises that ``what is the probability that the product of these two elements is \({\overline m}\)?'' We derive explicit formulas to compute the probability that the product of two elements chosen at random (with replacement) from \({\mathbb{Z}_n}\) is \({\overline m}\). Also we obtain some bounds for this probability.On the asymptotic behaviour of Hilbert functions of graded rings and modules: applications to the Hilbert-Samuel functions of well filtered modules, multiplicity and analytic spread of nice filtrations.https://zbmath.org/1449.130032021-01-08T12:24:00+00:00"Sangare, Daouda"https://zbmath.org/authors/?q=ai:sangare.daoudaSummary: This paper is devoted to the study of the asymptotic behaviour of Hilbert functions on finitely generated graded modules and particularly to some investigations on the asymptotic behaviour of Hilbert-Samuel functions on finitely generated well filtered modules. Consider a graded Noetherian ring \(A=\bigoplus_{n\in \mathbb{N}}A_n\) which may be assumed to be of the form \(A=A_0[x_1,\ldots ,x_s]\), where each element \(x_i\) is homogeneous and \(\deg x_i=k_i\ge 1\) for all \(i\). Suppose that the ring \(A_0\) is artinian. Let \(M =\bigoplus_{n\in \mathbb{Z}}M_n\) be a finitely generated positively \(\mathbb{Z} \)-graded \(A\)-module. Then the length \(l_{A_0}(M_n)\) of the \(A_0\)-module \(M_n\) is finite for all \(n\in \mathbb{Z} \). The Hilbert function \(H(M,-)\) of \(M\) is defined as \(H(M,n)= l_{A_0}(M_n)\) for all \(n\in \mathbb{Z} \). By a well known Hilbert Theorem, if \(k_i=1\) for all \(i\), then there exists a unique polynomial \(P(X)\in \mathbb{Q}[X]\) with \(\deg P(X)=d-1\) such that \(H(M,n)=P(n)\) for all large integers \(n\), where \(d\) denotes the Krull dimension of the \(A\)-module \(M\). But if at least one of the homogeneous generators \(x_i\) has degree \(k_i\ge 2,\) then such a polynomial need not exist. \textit{H. Dichi} and the author [J. Pure Appl. Algebra 138, No. 3, 205--213 (1999; Zbl 0934.13010)] had shown in that case that the Hilbert function \(H(M,-)\) of \(M\) has a good asymptotic behaviour. More precisely, they had shown that \(H(M,-)\) is a quasi-polynomial function. Then this result is applied, with some minor but necessary adaptation, to the graded \(G_f(A)\)-module \(G_{\Phi }(M)=\bigoplus_{n\ge 0}\frac{M_n}{M_{n+1}} \), where \(f=(I_n)_{n\in \mathbb{Z}}\) is a nice filtration of the ring \(A\) and where \(M\) is a finitely generated \(A\)-module which is no longer graded but which is endowed with a filtration \(\Phi =(M_n)_{n\in \mathbb{Z}}\) compatible with the filtration \(f\). Here \(G_f(A)=\bigoplus_{n\ge 0}\frac{I_n}{I_{n+1}}\) is the graded ring associated with \(f\). This leads in particular to the definition of a concept of multiplicity \(e_f(M)\) of \(M\) with respect to the filtration \(f\) which is a rational number. If the filtration \(f\) is non \(I\)-adic, then its multiplicity \(e_f(M)\) need not be an integer unlikely for the multiplicity of an ideal. So it does not have any geometrical interpretation. However, it is shown in [loc. cit.] that, for a given Noetherian filtration \(f=(I_n)_{n\in \mathbb{Z}}\) on the Noetherian finite dimensional ring \(A\), if \(l_A(\frac{M}{I_1M})<+\infty \), then the multiplicity function \(M\longmapsto e_f (M)\) behaves well on short exact sequences of finitely generated \(A\)-modules. As a second application of the Hilbert Theorem, we introduce a satisfactory concept of analytic spread for filtrations and we show that this analytic spread is of ``asymptotic nature'' for nice filtrations.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.A construction of Cohen-Macaulay graphs.https://zbmath.org/1449.130012021-01-08T12:24:00+00:00"Ahmad, Safyan"https://zbmath.org/authors/?q=ai:ahmad.safyan"Kanwal, Shamsa"https://zbmath.org/authors/?q=ai:kanwal.shamsaSummary: We present a technique to construct Cohen-Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen-Macaulay paths.On the \(h\)-vectors of the powers of graded ideals.https://zbmath.org/1449.130152021-01-08T12:24:00+00:00"Arkian, Seyed Shahab"https://zbmath.org/authors/?q=ai:arkian.seyed-shahab"Mafi, Amir"https://zbmath.org/authors/?q=ai:mafi.amir\(S = K[X_1,\dots,X_n]\) denotes a polynomial ring, over a field \(K\), equipped with the standard grading, and \(I \subset S\) is a graded ideal. The central aim of this paper is to study the postulation number of \(I^k\) (briefly \(\mathrm{psln}(I^k)\)) as a function of \(k\). A first result says that, if \(d\) is the largest degree of a set \((f_1,\dots,f_m), \mathrm{deg} (f_j) = d_j\), of generators of \(I\), there exists an integer \(c\) such that \(\mathrm{plsn}(I^k) \leq d k+c, \forall k\). The proof relais on a direct computation on the Rees algebra \(R(I) = \bigoplus I^k t^k\), considered as a finitely generated bigraded module over \(A = K[X_1,\dots,X_n, Y_1,\dots,Y_m]\), with bigrading defined by \(\mathrm{deg} (X_i) = (1,0), \mathrm{deg}(Y_j =( d_j,1)\). What comes after is based on the (already known) relation \(\mathrm{psln}(M) = \mathrm{deg} H_M(t) \), where \(M\) is any finitely generated graded module, generated in non negative degree, and \(H_M(t) = = \sum_j H(M,j)t^j = Q_M(t)/(1-t)^{\delta}\) is the Hilbert series of \(M\), written as a rational function, with \(\delta\) the Krull dimension of \(M\). Using a free resolution of \(I^k\), the authors prove the relation \(\mathrm{psln}(I^k) \leq \mathrm{reg} (I^k)-1\). This permits them to use some known properties of \(\mathrm{reg}(I^k)\) to show that if all generators of \(I\) are in degree \(d\), then there exists an integer \(c\) such that \(\mathrm{psln}(I^k) = dk+c, k\gg 0\). The last paragraph is devoted to the study of the higher itereted Hilbert coefficients of \(I^k\), in relation with the h-vector of \(I\). In particular, they prove that if \(I\) is generated in a single degree, then those coefficients are polynomials, for \(k\gg 0\).
Reviewer: Carla Massaza (Torino)The transfer ideal under the action of a nonmetacyclic group in the modular case.https://zbmath.org/1449.130062021-01-08T12:24:00+00:00"Jia, Panpan"https://zbmath.org/authors/?q=ai:jia.panpan"Nan, Jizhu"https://zbmath.org/authors/?q=ai:nan.ji-zhuSummary: Let \({F_q}\) be a finite field of characteristic \(p(p \ne 2)\) and \({V_4}\) a four-dimensional \({F_q}\)-vector space. In this paper, we mainly determine the structure of the transfer ideal for the ring of polynomials \({F_q}[{V_4}]\) under the action of a nonmetacyclic \(p\)-group \(P\) in the modular case. We prove that the height of the transfer ideal is 1 using the fixed point sets of the elements of order \(p\) in \(P\) and that the transfer ideal is a principal ideal.New relation formula for generating functions.https://zbmath.org/1449.130182021-01-08T12:24:00+00:00"Chammam, Wathek"https://zbmath.org/authors/?q=ai:chammam.wathekSummary: In this paper, we develop a new relation between certain types of generating functions using formal algorithmic methods. As an application, we give a relation between the generating function and finite-type relations between polynomial sequences.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 determination of when the fractions of \({Z_n}\) are fields.https://zbmath.org/1449.130092021-01-08T12:24:00+00:00"Zhuang, Ying"https://zbmath.org/authors/?q=ai:zhuang.ying"Zhang, Xiaojin"https://zbmath.org/authors/?q=ai:zhang.xiaojin"Zhu, Ziyang"https://zbmath.org/authors/?q=ai:zhu.ziyangSummary: By using the adjoint relation of zero divisors in the ring \({Z_n}\), we give a criterion of when the fraction of \({Z_n}\) can be a field. More precisely, we prove that the fraction of \({Z_n}\) can be a field if and only if there is a prime number \(p\) such that \(p\) is a divisor of \(n\) and \({p^2}\) is not a divisor of \(n\).On the maximal spectrum of lattice modules.https://zbmath.org/1449.060282021-01-08T12:24:00+00:00"Phadatare, Narayan"https://zbmath.org/authors/?q=ai:phadatare.narayan"Kharat, Vilas"https://zbmath.org/authors/?q=ai:kharat.vilas-s"Ballal, Sachin"https://zbmath.org/authors/?q=ai:ballal.sachinSummary: Let \(M\) be a lattice module over a \(C\)-lattice \(L\). The maximal spectrum \(\mathrm{Max}(M)\) of \(M\) is the collection of all maximal elements of \(M\). In this paper, we study the topology on \(\mathrm{Max}(M)\) and also establish the interrelations between the topological properties of \(\mathrm{Max}(M)\) and the algebraic properties of \(M\).Light dual multinets of order six in the projective plane.https://zbmath.org/1449.050392021-01-08T12:24:00+00:00"Bogya, N."https://zbmath.org/authors/?q=ai:bogya.n"Nagy, G. P."https://zbmath.org/authors/?q=ai:nagy.gabor-peterLet \(\mathbb{K}\) be a field, \(Q\) a quasigroup, and for \(i = 1, 2, 3\), let \(\alpha_i : Q \rightarrow \mathrm{PG}(2,\mathbb{K})\) be maps such that the points \(\alpha_1(x), \alpha_2(y)\) and \(\alpha_3(x \cdot y)\) are collinear for all \(x, y \in Q\). Define the multisets \(\Lambda_i = \alpha_i(Q), \ i = 1, 2, 3\). Then \((\Lambda_1, \Lambda_2, \Lambda_3)\) is a dual multinet, labeled by \(Q\). If the maps \(\alpha_i\) are injective and their images \(\Lambda_i\) are disjoint, then the dual multinet is called light. If a line \(\ell\) intersects two components \(\Lambda_i\), \(\Lambda_j\) then there is an integer \(r\) such that \(r = |\ell \cap \Lambda_1| = |\ell \cap \Lambda_2| = |\ell \cap \Lambda_3|\); this integer \(r\) is called the length of \(\ell\) with respect to \((\Lambda_1, \Lambda_2, \Lambda_3)\). According to the authors, previous work suggests that ``the length \(r>1\) of lines of the light dual multinet makes a big difference in their geometric structure. While for \(r \geq 9\), the light dual multinet is well structured in (the) geometric and algebraic sense, the case of small \(r\), especially \(r=2\) shows many irregularities''. In this paper, the authors classify all abstract light dual multinets of order 6 with a unique line of length \(r>1\), and they compute all possible realizations of these abstract light dual multinets in projective planes over fields of characteristic zero.
Reviewer: Juan C. Migliore (Notre Dame)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)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 \({A_n}\).Some properties of the maximal graph of a commutative ring.https://zbmath.org/1449.051342021-01-08T12:24:00+00:00"Mahmudi, F."https://zbmath.org/authors/?q=ai:mahmudi.fatemeh"Soleimani, M."https://zbmath.org/authors/?q=ai:soleimani.manuchehr|soleimani.mohammad|soleimani.meisam|soleimani.m-j|soleimani.majid"Naderi, M. H."https://zbmath.org/authors/?q=ai:naderi.mohammad-hossein|naderi.mohamad-hosein|naderi.mohammad-hassanSummary: Let \(R\) be a commutative ring with identity. The maximal graph of \(R\) is the graph whose vertices are elements of \(R\), where two distinct vertices \(x\) and \(y\) are adjacent if and only if there is a maximal ideal of \(R\) containing both. In this paper we investigate some properties of two subgraphs of this graph.Clean exactness and derived categories.https://zbmath.org/1449.130142021-01-08T12:24:00+00:00"Li, Ruiting"https://zbmath.org/authors/?q=ai:li.ruiting"Yang, Gang"https://zbmath.org/authors/?q=ai:yang.gang"Wang, Xiaoqing"https://zbmath.org/authors/?q=ai:wang.xiaoqingSummary: Over a commutative ring, the notions of clean exactness and clean derived categories are introduced, the equivalent characterizations of clean exactness for short exact sequences and exact complexes are given, and the properties of clean derived categories are investigated. In particular, it is proved that bounded clean derived categories can be realized as certain homotopy categories.The square mapping graph of \(2\times 2\) matrix rings over prime fields.https://zbmath.org/1449.051392021-01-08T12:24:00+00:00"Tang, Gaohua"https://zbmath.org/authors/?q=ai:tang.gaohua"Zhang, Hengbin"https://zbmath.org/authors/?q=ai:zhang.hengbin"Wei, Yangjiang"https://zbmath.org/authors/?q=ai:wei.yangjiangSummary: Let \(R\) be a ring with identity. The square mapping graph of \(R\) is a digraph \(\Gamma(R)\) defined on the elements of \(R\) and with an edge from a vertex a to \(a\) vertex \(b\) if and only if \(a^2=b\). Let \(\mathbb{M}_2(\mathbb{Z}_p)\) be the \(2\times 2\) matrix ring over the field \(\mathbb{Z}_p\), where \(p\) is prime. In this paper, we completely determine the structure of \(\Gamma(\mathbb{M}_2(\mathbb{Z}_p))\) by showing its two disjoint induced subgraphs.On zero divisor graphs of the rings \(Z_n\).https://zbmath.org/1449.130082021-01-08T12:24:00+00:00"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Aijaz, M."https://zbmath.org/authors/?q=ai:aijaz.malla"Bhat, M. Imran"https://zbmath.org/authors/?q=ai:bhat.m-imranSummary: For a commutative ring \(R\) with non-zero zero-divisor set \(Z^*(R)\), the zero-divisor graph of \(R\) is \(\varGamma (R)\) with vertex set \(Z^*(R)\), where two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). The zero-divisor graph structure of \(\mathbb{Z}_{p^n}\) is described. We determine the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-divisor graph of \(\mathbb{Z}_{p^n}\). Further, we provide a partition of the vertex set of \(\varGamma (\mathbb{Z}_{p^n})\) into distance similar equivalence classes and we show that in this graph the upper dimension equals the metric dimension. Also, we discuss similar properties of the compressed zero-divisor graph.