Recent zbMATH articles in MSC 13Chttps://www.zbmath.org/atom/cc/13C2021-06-15T18:09:00+00:00WerkzeugOn the Gauss algebra of toric algebras.https://www.zbmath.org/1460.130202021-06-15T18:09:00+00:00"Herzog, Jürgen"https://www.zbmath.org/authors/?q=ai:herzog.jurgen"Jafari, Raheleh"https://www.zbmath.org/authors/?q=ai:jafari.raheleh"Nasrollah Nejad, Abbas"https://www.zbmath.org/authors/?q=ai:nasrollah-nejad.abbasLet \(A\) be a \(K\)-subalgebra of the polynomial ring \(S = K[x_{1},\dots, x_{d} ]\) of dimension \(d\), generated by finitely many monomials of degree \(r\) . Then, the Gauss algebra \(G(A)\) of \(A\) is generated by monomials of degree \((r-1)d\) in \(S\). The authors describe the generators and the structure of \(G(A)\), when \(A\) is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph \(G\) with one loop, the embedding dimension of \(G(A)\) is bounded by
the complexity of the graph \(G\).
Reviewer: Mohammad Javad Nikmehr (Tehran)Finiteness of the number of minimal atoms in Grothendieck categories.https://www.zbmath.org/1460.180052021-06-15T18:09:00+00:00"Kanda, Ryo"https://www.zbmath.org/authors/?q=ai:kanda.ryoSummary: For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object.Almost complete intersection binomial edge ideals and their Rees algebras.https://www.zbmath.org/1460.130412021-06-15T18:09:00+00:00"Jayanthan, A. V."https://www.zbmath.org/authors/?q=ai:jayanthan.a-v"Kumar, Arvind"https://www.zbmath.org/authors/?q=ai:kumar.arvind"Sarkar, Rajib"https://www.zbmath.org/authors/?q=ai:sarkar.rajibLet \(G\) be a graph on \(n\) vertices and \(J_G\) be the binomial edge ideal in the polynomial ring \(S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]\), where \(K\) is a field. The authors study the graphs whose binomial edge ideals are almost complete intersection. They show that if \(G\) is a tree and not a path, then \(J_G\) is an almost complete intersection iff \(G\) is obtained by adding an edge between two vertices of two paths and if \(G\) is connected and not a tree, then \(J_G\) is an almost complete intersection iff \(G\) is obtained by adding an edge between two vertices of a path or by adding a path to each vertex of \(C_3\). It is also shown that if \(J_G\) is almost complete intersection, then the form ring \(gr_S(J_G)\) and the Rees ring \(R_S(J_G)\) are both Cohen-Macaulay.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Regularity and the Gorenstein property of \(L\)-convex polyominoes.https://www.zbmath.org/1460.052052021-06-15T18:09:00+00:00"Ene, Viviana"https://www.zbmath.org/authors/?q=ai:ene.viviana"Herzog, Jürgen"https://www.zbmath.org/authors/?q=ai:herzog.jurgen"Qureshi, Ayesha Asloob"https://www.zbmath.org/authors/?q=ai:qureshi.ayesha-asloob"Romeo, Francesco"https://www.zbmath.org/authors/?q=ai:romeo.francescoSummary: We study the coordinate ring of an \(L\)-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein \(L\)-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen-Macaulay type of any \(L\)-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.Some remarks on locally perfect rings.https://www.zbmath.org/1460.130082021-06-15T18:09:00+00:00"Zhou, Dechuan"https://www.zbmath.org/authors/?q=ai:zhou.dechuan"Kim, Hwankoo"https://www.zbmath.org/authors/?q=ai:kim.hwankoo"Zhang, Xiaolei"https://www.zbmath.org/authors/?q=ai:zhang.xiaolei"Hu, Kui"https://www.zbmath.org/authors/?q=ai:hu.kuiNearly semi-2-absorbing submodules and related concepts.https://www.zbmath.org/1460.130282021-06-15T18:09:00+00:00"Mohammadali, Haibat K."https://www.zbmath.org/authors/?q=ai:mohammadali.haibat-k"Mohammed, Akram S."https://www.zbmath.org/authors/?q=ai:mohammed.akram-sSummary: In this article, \(R\) is commutative ring with identity and \(Y\) is a left unitary \(R\)-module. A proper submodule \(L\) of \(Y\) is called nearly semiprime submodule if whenever \(r^ny\in L\), where \(r\in R\) and \(y\in Y\), \(n\in Z^+\), implies that \(ry\in L+J(Y)\), where \(J(Y)\) is the Jacobson radical of \(Y\). This concept in courage us to introduce the concept nearly semi-2-absorbing submodule as a generalization of nearly semiprime submodule, where a proper submodule \(L\) of \(Y\) is called nearly semi-2-absorbing submodule of \(Y\) if whenever \(a^2y\in L\), where \(a\in R\), \(y\in Y\), implies that either \(ay\in L+J(Y)\) or \(a^2\in [L:Y]\). Many basic properties, and characterization of this concept are introduce. On the other hand the relation of this concept with other classes of modules are studied.On the \(k\)-torsion of the module of differentials of order \(n\) of hypersurfaces.https://www.zbmath.org/1460.130472021-06-15T18:09:00+00:00"de Alba, Hernán"https://www.zbmath.org/authors/?q=ai:de-alba.hernan"Duarte, Daniel"https://www.zbmath.org/authors/?q=ai:duarte.daniel-c-sSummary: We characterize the \(k\)-torsion freeness of the module of differentials of order \(n\) of a point of a hypersurface in terms of the singular locus of the corresponding local ring.Endoregular modules.https://www.zbmath.org/1460.130172021-06-15T18:09:00+00:00"Anderson, D. D."https://www.zbmath.org/authors/?q=ai:anderson.daniel-d"Juett, J. R."https://www.zbmath.org/authors/?q=ai:juett.jason-rThe authors investigate endoregular modules, that is, modules whose endomorphism rings satisfy various conditions related to von Neumann regularity. A ring \(R\) is called (one-sided) (unit-)regular if for each \(x \in R\) there is a (one-sided) (unit) \(y \in R\) with \(x = xyx\). A ring \(R\) is strongly regular if it is regular and abelian (i.e., idempotents are central), or equivalently for each \(x\in R\) there is a \(y\in R\) with \(x = x^2y\). Thus, a module is called (one-sided) (unit-)endoregular (resp., strongly endoregular, commutative endoregular) if its endomorphism ring is (one-sided) (unit-)regular (resp., strongly regular, commutative regular). The general relationship between these properties is: commutative endoregular \(\Rightarrow\) strongly endoregular \(\Rightarrow\) unit-endoregular \(\Rightarrow\) one-sided unit-endoregular \(\Rightarrow\) endoregular and none can be reversed. The authors view the purpose of the article as threefold: (1) Expand the general theory of these various forms of endoregularity (2) extend known results about endoregularity in abelian groups to modules over one-dimensional commutative rings with Noetherian spectrum and (3) generalize endoregular modules (over commutative rings) and then investigate their properties and characterize these modules. The paper contains a good introduction and extensive bibliography about the historical development of research on these topics. The article is well written and efficiently presents a large number of interesting results. There are many examples provided throughout to illustrate these properties as well as demonstrating when implications do not hold between properties.
Reviewer: Christopher P. Mooney (Fulton)The Zariski topology-graph of modules over commutative rings. II.https://www.zbmath.org/1460.130192021-06-15T18:09:00+00:00"Ansari-Toroghy, H."https://www.zbmath.org/authors/?q=ai:ansari-toroghy.habibollah"Habibi, S."https://www.zbmath.org/authors/?q=ai:habibi.shokoofe|habibi.shokoufehSummary: Let \(M\) be a module over a commutative ring \(R\). In this paper, we continue our study about the Zariski topology-graph \(G(\tau_T)\) which was introduced in [the authors, Commun. Algebra 42, No. 8, 3283--3296 (2014; Zbl 1295.13016)]. For a non-empty subset \(T\) of \(\text{Spec}(M)\), we obtain useful characterizations for those modules \(M\) for which \(G(\tau_T)\) is a bipartite graph. Also, we prove that if \(G(\tau_T)\) is a tree, then \(G(\tau_T)\) is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between \(\chi (G(\tau_T))\) and \(\omega (G(\tau_T))\).On weakly 2-irreducible ideals of commutative rings.https://www.zbmath.org/1460.130072021-06-15T18:09:00+00:00"Zeidi, Nabil"https://www.zbmath.org/authors/?q=ai:zeidi.nabilSummary: All rings are commutative with \(1 \neq 0\). The purpose of this paper is to investigate the concept of weakly 2-irreducible ideals generalizing weakly irreducible ideals and strongly 2-irreducible ideals. We say that a proper ideal \(I\) of a ring \(R\) is weakly 2-irreducible provided that for each ideals \(J\), \(K\) and \(L\) of \(R\), \(J \cap K \cap L \subseteq I\) implies that either \(J \cap K \subseteq \sqrt{I}\) or \(J \cap L \subseteq \sqrt{I}\) or \(K \cap L \subseteq \sqrt{I}\). A number of results concerning weakly 2-irreducible ideals are given. For instance, the relationships between the notions weakly 2-irreducible, 2-absorbing, 2-absorbing primary and 2-absorbing quasi-primary in different rings, has been given.Zariski locality of quasi-coherent sheaves associated with tilting.https://www.zbmath.org/1460.140452021-06-15T18:09:00+00:00"Hrbek, Michal"https://www.zbmath.org/authors/?q=ai:hrbek.michal"Stovícek, Jan"https://www.zbmath.org/authors/?q=ai:stovicek.jan"Trlifaj, Jan"https://www.zbmath.org/authors/?q=ai:trlifaj.jan-junSummary: A classic result by \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)] says that the notion of an (infinite-dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for \(n=0)\) of the locality of more general notions of quasi-coherent sheaves related to (infinite-dimensional) \(n\)-tilting modules and classes. Here, we prove the latter locality for all \(n\) and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is Noetherian).The Euler-Poincaré characteristic of joint reductions and mixed multiplicities.https://www.zbmath.org/1460.130452021-06-15T18:09:00+00:00"Thanh, Truong Thi Hong"https://www.zbmath.org/authors/?q=ai:thanh.truong-thi-hong"Viet, Duong Quoc"https://www.zbmath.org/authors/?q=ai:duong-quoc-viet.On the Betti numbers of edge ideals of skew Ferrers graphs.https://www.zbmath.org/1460.130352021-06-15T18:09:00+00:00"Hoang, Do Trong"https://www.zbmath.org/authors/?q=ai:hoang.do-trongA Ferrers diagram \(D_{X,Y}\) with \(\lambda = (\lambda_1 = m \geq \cdots \geq \lambda_n)\) on \(X = \{x_1, \ldots, x_n\}\) and \(Y = \{y_1, \ldots, y_m\}\) is defined as an array of cells doubly indexed by pairs \((x_i, y_j)\) with \(1 \leq i \leq n\), \(m + 1 - \lambda_i \leq j \leq m.\) The difference between two Ferrers diagrams is called a skew Ferrers diagram. In other words, the skew Ferrers diagram \(D_{X,Y}\) on \((X, Y)\) is defined by two nonincreasing sequences of integers, \(\lambda = (\lambda_1 = m \geq \cdots \geq \lambda_n)\) and \(\mu = (\mu_1 \geq \cdots \geq \mu_n)\) and \(\lambda_i \geq \mu_i\) for all \(i.\) More precisely, its cells are \(\{(x_i, y_j) \mid 1 \leq i \leq n,m+ 1 - \lambda_i \leq j \leq m- \mu_i\}.\) A skew Ferrers diagram is called staircase if \(m = n\), \(\lambda = (n, n -1, \ldots, 1)\), and \(\mu = (s, s- 1, \ldots, 1, 0, \ldots, 0),\) where \(s \leq n-2.\) A bipartite graph \(G\) on two distinct vertex sets \(X\) and \(Y\) is called a Ferrers graph (respectively (staircase) skew Ferrers graph) if \(\{x_i, y_j\}\) is an edge of \(G\) whenever \((x_i, y_j)\) is a cell in the Ferrers diagram (respectively (staircase) skew Ferrers diagram).
Let \(I(G)\) denote the edge ideal of a simple graph \(G.\) In the paper under review, the author studies the vanishing properties of Betti numbers of the last column of Betti table of \(I(G).\) A nonzero graded Betti number \(\beta_{i,j}(I)\) is called an extremal Betti number if \(\beta_{r,s}(I) = 0\) for all pairs \((r, s) \neq (i, j)\) with \(r \geq i\) and \(s \geq j.\) Observe that the extremal Betti number is unique if and only if \(\beta_{p,p+r}(I) \neq 0\), where \(p = \text{pd}(I)\) is the projective dimension and \(r = \text{reg}(I)\) is the regularity of \(I\). The author proves the following:
Theorem. Let \(G\) be a staircase skew Ferrers graph. Then \(\beta_p(I(G)) = \beta_{p,p+r}(I(G)).\)
Let \(J_G\) denote the binomial edge ideal of a simple graph \(G.\) It is known that \(\beta_{i,j}(J_G) \leq \beta_{i,j}(\text{in}(J_G))\) for all \(i, j\), where \(\text{in}(J_G)\) is the initial ideal with respect to the lexicographic order. Also, \(J_G\) has a quadratic Gröbner basis with respect to the above order if and only if the graph \(G\) is closed with respect to the given labeling. In other words, if \(G\) satisfies the following condition: whenever \(\{i, j\}\) and \(\{i, k\}\) are edges of \(G\) and either \(i < j\), \(i < k\) or \(i > j\), \(i > k\) then \(\{j, k\}\) is also an edge of \(G.\) One calls a graph \(G\) closed if it is closed with respect to some labeling of its vertices. When \(G\) is a closed graph, in [\textit{V. Ene} et al., Nagoya Math. J. 204, 57--68 (2011; Zbl 1236.13011)] it was conjectured that \(\beta_{i,j}(J_G) = \beta_{i,j}(\text{in}(J_G))\) for all \(i, j.\) The conjecture is proved in the affirmative for Cohen-Macaulay binomial edge ideals in [loc. cit.], and for closed graphs which contain at most two cliques in [\textit{H. Baskoroputro}, ``On the binomial edge ideals of proper interval graphs'', Preprint, \url{arXiv:1611.10117}]. Herzog and Rinaldo also considered the conjecture for the extremal Betti numbers of binomial edge ideals of block graphs in [\textit{J. Herzog} and \textit{G. Rinaldo}, Electron. J. Comb. 25, No. 1, Research Paper P1.63, 10 p. (2018; Zbl 1395.13010)]. The author proves the following result in the paper:
Theorem. If \(G\) is a staircase closed graph, then \(\text{reg}(J_G) = \text{reg}(\text{in}(J_G)) =: r,\) \(\text{pd}(J_G) = \text{pd}(\text{in}(J_G)) =: p\), and \(\beta_p(J_G) = \beta_{p,p+r}(J_G) = \beta_p(\text{in}(J_G)) = \beta_{p,p+r}(\text{in}(J_G)) \neq 0.\)
In the last section, the author gives a formula for the unique extremal Betti numbers of binomial edge ideals of closed graphs.
Theorem. Let \(G\) be a closed graph without cut vertices with \(\mu(G) = (\mu_1, \ldots, \mu_n) \in \mathbb{N}^n.\) Then \(\text{reg}(J_G) = 3\) if and only if \(\mu_1 = \cdots = \mu_s =: \mu \geq 1\) and \(s \geq 1\), where \(s := \text{min}\{k-1 \mid \mu k = 0\}.\) In particular, \(\beta_p(J_G) = \beta_{p,p+3}(J_G) = \beta_p(\text{in}(J_G)) = \beta_{p,p+3}(\text{in}(J_G)) = s \mu,\) where \(p = \text{pd}(J_G) = \text{pd}(\text{in}(J_G)) = 2n-\mu- s- 3.\)
The author also mentions that he learned later that the conjecture on the equality of the extremal Betti numbers of \(J_G\) and \(\text{in}(J_G)\) has been completely solved in [\textit{A. Conca} and \textit{M. Varbaro}, Invent. Math. 221, No. 3, 713--730 (2020; Zbl 1451.13076)].
Reviewer: Kriti Goel (Mumbai)Second multiplication modules.https://www.zbmath.org/1460.130182021-06-15T18:09:00+00:00"Ansari-Toroghy, H."https://www.zbmath.org/authors/?q=ai:ansari-toroghy.habibollah"Farshadifar, F."https://www.zbmath.org/authors/?q=ai:farshadifar.faranakSummary: In this paper, we introduce second multiplication modules and obtain some related results. Also, we provide a counterexample to a previously published result concerning multiplication modules.Primality in semigroup rings.https://www.zbmath.org/1460.130042021-06-15T18:09:00+00:00"Boulayat, Brahim"https://www.zbmath.org/authors/?q=ai:boulayat.brahim"El Baghdadi, Said"https://www.zbmath.org/authors/?q=ai:el-baghdadi.saidLet \(S=(S,\ast ,e,\leq )\) be a cancellative divisibility (affords \(a\leq b\Leftrightarrow\) there is \(c\in S\) with \(a\ast c=b\)) monoid. An element \(s\) in \(S\) may be called primal if whenever \(s\leq a\ast b\) for \(a,b\in S\) we can write \(s=s_{1}\ast s_{2}\), with \(s_{i}\in S\) such that \(s_{1}\leq a\) and \(s_{2}\leq b\). We may call \(c\in S\) completely primal if each \(x\in S\) with \(x\leq c\) is primal. (If \(S\) is additively written (\(\ast =+\)), whether it is Abelian or non-Abelian, \(e\) becomes \(0\) and if \(S\) is multiplicatively written (\(\ast =\cdot\)) \(e\) becomes \(1\) and \(\leq\) becomes \(|\) (divides).) Call \(S\) a pre-Schreier monoid if \(s\) is primal for each \(s\in S\) and call \(S\) integrally closed if \(nx\geq ny\) for some positive integer \(n\) implies \(x\geq y\) for \(x,y\in S\). Because \(S\) is a cancellative and divisibility monoid, \(S\) can be embedded in a p.o. group \(G(S)\) with \(G(S)^{+}=S\) and if \(S\) is a pre-Schreier monoid, then \(G(S)\) is a Riesz group, a directed group that satisfies the Riesz interpolation property: given that \(x_{1},x_{2},\dots,x_{m};y_{1},y_{2},\dots,y_{n}\in G\) such that \(x_{i}\leq y_{j}\) for all \(i\in \lbrack 1,m],j\in \lbrack 1,n]\) there is \(z\in G\) such that \(x_{i}\leq z\leq y_{j}\) for all \((i,j)\in \lbrack 1,m]\times \lbrack 1,n])\), according to (4) of Theorem 2.2 of \textit{L. Fuchs} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19, 1--34 (1965; Zbl 0125.28703)]. \textit{P. M. Cohn} [Proc. Camb. Philos. Soc. 64, 251--264 (1968; Zbl 0157.08401)] called an integrally closed integral domain \(D\) a Schreier domain if \((D\backslash \{0\},\cdot ,|,1)\) was pre-Schreier. The reviewer gave the name pre-Schreier to a domain \(D\) if \((D\backslash \{0\},\cdot ,|,1)\) was pre-Schreier and indicated in [\textit{M. Zafrullah}, Commun. Algebra 15, 1895--1920 (1987; Zbl 0634.13004)] that pre-Schreier domains had a life of their own.
The authors of the paper under review explore (complete) primality (the quality of being (completely) primal) in the set up of monoids and monoid domains \(D[X;\Gamma]=\{\sum d_{\gamma}X^{\gamma}\mid d_{\gamma}\in D,\gamma \in \Gamma,\text{ with almost all }d_{\gamma}=0\}\). They do it by adapting known results on (complete) primality in integral domains and monoids to their preferred set up, by occasionally simplifying the known proofs. However, as expected, they end up writing a sort of survey on when \(D[X,\Gamma]\) is a Schreier domain. Of particular interest to them is the monoid ring \(D[X;\Gamma]\) that is an integral domain. According to Theorem 8.1 of \textit{R. Gilmer}'s book [Commutative semigroup rings. Chicago-London: The University of Chicago Press (1984; Zbl 0566.20050)] that \(D[X;\Gamma]\) is a domain if and only if \(D\) is a domain and \(\Gamma\) is a grading (additive, cancellative and torsionfree) monoid. \textit{R. Matsuda} opened the field, from Schreier angle, by proving several interesting results in [Comment. Math. Univ. St. Pauli 33, 79--86 (1984; Zbl 0558.13004)] and in [Math. J. Okayama Univ. 39, 41--44 (1997; Zbl 0937.20042)]. The upshot of his work was that for an integrally closed \(D\) the ring \(D[X;\Gamma]\) is a Schreier domain if and only if \(D\) is a Schreier domain and \(\Gamma\) is an integrally closed monoid satisfying a certain condition. The activity came to a sort of close when \textit{G. Brookfield} and \textit{D. E. Rush} [J. Pure Appl. Algebra 195, No. 3, 225--230 (2005; Zbl 1099.13037)] took it from the graded domain end and showed that \(D[X;\Gamma]\) is pre-Schreier if and only if \(D[X;\Gamma]\) is Schreier.
For the entire collection see [Zbl 1446.20006].
Reviewer: Muhammad Zafrullah (Pocatello)