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1. CMB 2017 (vol 61 pp. 865)

Shaul, Liran
 Homological Dimensions of Local (Co)homology over Commutative DG-rings Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$ is a noetherian ring, and for each $i\lt 0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension. Keywords:local cohomology, derived completion, homological dimension, commutative DG-ringCategories:13B35, 13D05, 13D45, 16E45

2. CMB 2017 (vol 61 pp. 346)

Polak, Jason K. C.
 Counting Separable Polynomials in $\mathbb{Z}/n[x]$ For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For instance, we show that the number of separable polynomials in $\mathbb{Z}/n[x]$ that are separable is $\phi(n)n^d\prod_i(1-p_i^{-d})$ where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $\phi$ is Euler's totient function. Keywords:separable algebra, separable polynomialCategories:13H05, 13B25, 13M10

3. CMB 2017 (vol 60 pp. 225)

Bahmanpour, Kamal; Naghipour, Reza
 Faltings' Finiteness Dimension of Local Cohomology Modules Over Local Cohen-Macaulay Rings Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$ and if $\operatorname{mAss}_R(R/I)$ is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring. Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomologyCategories:13D45, 14B15

4. CMB 2017 (vol 61 pp. 166)

Miranda-Neto, Cleto B.
 A module-theoretic characterization of algebraic hypersurfaces In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor. Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisorCategories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

5. CMB 2016 (vol 60 pp. 3)

Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud
 The Co-annihilating-ideal Graphs of Commutative Rings Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by $\mathcal{A}_R$, is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties. Keywords:commutative ring, co-annihilating ideal graphCategories:13A15, 16N40

6. CMB 2016 (vol 60 pp. 63)

Chang, Gyu Whan
 Power Series Rings Over PrÃ¼fer $v$-multiplication Domains, II Let $D$ be an integral domain, $X^1(D)$ be the set of height-one prime ideals of $D$, $\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be two disjoint nonempty sets of indeterminates over $D$, $D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and $D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type power series ring over $D[\{X_{\beta}\}]$. Assume that $D$ is a PrÃ¼fer $v$-multiplication domain (P$v$MD) in which each proper integral $t$-ideal has only finitely many minimal prime ideals (e.g., $t$-SFT P$v$MDs, valuation domains, rings of Krull type). Among other things, we show that if $X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$, then ${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D - \{0\}}$ is a Krull domain. We also prove that if $D$ is a $t$-SFT P$v$MD, then the complete integral closure of $D$ is a Krull domain and ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every height-one maximal $t$-ideal $M$ of $D$. Keywords:Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ringCategories:13A15, 13F05, 13F25

7. CMB 2016 (vol 59 pp. 794)

Hashemi, Ebrahim; Amirjan, R.
 Zero-divisor Graphs of Ore Extensions over Reversible Rings Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$, when $R$ is reversible and $(\alpha,\delta)$-compatible. Keywords:zero-divisor graphs, reversible rings, McCoy rings, polynomial rings, power series ringsCategories:13B25, 05C12, 16S36

8. CMB 2016 (vol 59 pp. 617)

Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
 Canonical Systems of Basic Invariants for Unitary Reflection Groups It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group. Keywords:basic invariant, invariant theory, finite unitary reflection groupCategories:13A50, 20F55

9. CMB 2016 (vol 59 pp. 641)

 Some Results on the Annihilating-ideal Graphs The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here, we show that if $R$ is a reduced ring and the independence number of $\mathbb{AG}(R)$ is finite, then the edge chromatic number of $\mathbb{AG}(R)$ equals its maximum degree and this number equals $2^{|{\rm Min}(R)|-1}-1$; also, it is proved that the independence number of $\mathbb{AG}(R)$ equals $2^{|{\rm Min}(R)|-1}$, where ${\rm Min}(R)$ denotes the set of minimal prime ideals of $R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph $\mathbb{AG}(R)$ is not Eulerian, and it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand. Keywords:annihilating-ideal graph, independence number, edge chromatic number, bipartite, cycleCategories:05C15, 05C69, 13E05, 13E10

10. CMB 2016 (vol 59 pp. 748)

Dolžan, David
 The Metric Dimension of the Total Graph of a Finite Commutative Ring We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields. Keywords:total graph, finite ring, metric dimensionCategories:13M99, 05E40

11. CMB 2016 (vol 59 pp. 271)

 Artinianness of Composed Graded Local Cohomology Modules Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with local base ring $(R_{0}, \mathfrak{m}_{0})$ and let $R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated graded $R$-modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M, N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and $j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal of $R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$-primary ideal. Keywords:generalized local cohomology, Artinian, graded moduleCategories:13D45, 13E10, 16W50

12. CMB 2016 (vol 59 pp. 403)

Zargar, Majid Rahro; Zakeri, Hossein
 On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\mathfrak{d}_R X$, $\operatorname{\mathsf{Gfd}}_R X$ and $\operatorname{\mathsf{G_C-fd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$-module such that $\operatorname{H}_{\mathfrak{a}}^i(M)=0$ for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then $\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings. Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing moduleCategories:13D05, 13D45, 18G20

13. CMB 2015 (vol 59 pp. 197)

Rajaee, Saeed
 Quasi-copure Submodules All rings are commutative with identity and all modules are unital. In this paper we introduce the concept of quasi-copure submodule of a multiplication $R$-module $M$ and will give some results of them. We give some properties of tensor product of finitely generated faithful multiplication modules. Keywords:multiplication module, arithmetical ring, copure submodule, radical of submodulesCategories:13A15, 13C05, 13C13, , 13C99

14. CMB 2015 (vol 58 pp. 704)

Benamar, H.; Chandoul, A.; Mkaouar, M.
 On the Continued Fraction Expansion of Fixed Period in Finite Fields The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\in \mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\in \mathbb{F}_q[X]$ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $Per \sqrt {Q}=t$. Keywords:continued fractions, polynomials, formal power seriesCategories:11A55, 13J05

15. CMB 2015 (vol 58 pp. 664)

Vahidi, Alireza
 Betti Numbers and Flat Dimensions of Local Cohomology Modules Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$--module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$. Keywords:Betti numbers, flat dimensions, local cohomology modulesCategories:13D45, 13D05

16. CMB 2015 (vol 58 pp. 449)

Boynton, Jason Greene; Coykendall, Jim
 On the Graph of Divisibility of an Integral Domain It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity. Keywords:atomic, factorization, divisibilityCategories:13F15, 13A05

17. CMB 2015 (vol 58 pp. 320)

Llamas, Aurora; Martínez-Bernal, José
 Cover Product and Betti Polynomial of Graphs For disjoint graphs $G$ and $H$, with fixed vertex covers $C(G)$ and $C(H)$, their cover product is the graph $G \circledast H$ with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in C(G), j\in C(H)\}$. We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer $k$ there exists a connected bipartite graph $G$ such that $\operatorname{reg} R/I(G)=\mu_S(G)+k$, where, $I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$ is the Castelnuovo--Mumford regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong matching number of $G$; (ii) The graded Betti numbers of the complement of a tree only depends upon its number of vertices; (iii) The $h$-vector of $R/I(G\circledast H)$ is described in terms of the $h$-vectors of $R/I(G)$ and $R/I(H)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs. Keywords:Castelnuovo--Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial idealCategories:13D02, 05E45

18. CMB 2015 (vol 58 pp. 393)

Tang, Zhongming
 On Stanley Depths of Certain Monomial Factor Algebras Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$-variables over a field $K$ and $I$ a monomial ideal of $S$. According to one standard primary decomposition of $I$, we get a Stanley decomposition of the monomial factor algebra $S/I$. Using this Stanley decomposition, one can estimate the Stanley depth of $S/I$. It is proved that ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds for $S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$. Keywords:monomial ideal, size, Stanley depthCategories:13F20, 13C15

19. CMB 2014 (vol 58 pp. 134)

Nasseh, Saeed
 On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions We show under some conditions that a Gorenstein ring $R$ satisfies the Generalized Auslander-Reiten Conjecture if and only if so does $R[x]$. When $R$ is a local ring we prove the same result for some localizations of $R[x]$. Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein ringsCategories:13D07, 16E30, 16E65

20. CMB 2014 (vol 57 pp. 573)

Kiani, Sima; Maimani, Hamid Reza; Nikandish, Reza
 Some Results on the Domination Number of a Zero-divisor Graph In this paper, we investigate the domination, total domination and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$, where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied. Keywords:zero-divisor graph, domination numberCategories:05C75, 13H10

21. CMB 2014 (vol 57 pp. 477)

Eghbali, Majid
 On Set Theoretically and Cohomologically Complete Intersection Ideals Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities $\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq \operatorname{ara}(\mathfrak a) \leq l(\mathfrak a) \leq \mu(\mathfrak a)$ are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities. Keywords:set-theoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of idealsCategories:13D45, 13C14

22. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
 On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$? Keywords:uniserial module, Lie algebra, associative algebra, primitive elementCategories:17B10, 13C05, 12F10, 12E20

23. CMB 2013 (vol 57 pp. 413)

Samei, Karim
 On the Comaximal Graph of a Commutative Ring Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined a graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra + Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of $\Gamma(R)$ which consists of non-unit elements. We investigate the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$, where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of $\Gamma_2(R)$ and the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles and dominating sets are investigated and the algebraic and the topological characterizations are given for graphical properties of these graphs. Keywords:comaximal, Diameter, girth, cycles, dominating setCategory:13A99

24. CMB 2013 (vol 57 pp. 188)

 A Characterization of Bipartite Zero-divisor Graphs In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings $R$ with $1$, such that $R$ is finite or $|Nil(R)|\neq2$. Keywords:zero-divisor graph, bipartite graphCategories:13AXX, 05C25
 Strongly $0$-dimensional Modules In a multiplication module, prime submodules have the property, if a prime submodule contains a finite intersection of submodules then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodule. A multiplication module in which every prime submodule is strongly prime will be called strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules. Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular moduleCategories:13C99, 16D10