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The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x C y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian.

A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given.

Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G(R) are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G(R) are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)

A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given. 2010 Mathematics Subject Classification. Primary 05C15, 05C69, 05C17; Secondary 13M05.

The ring Z2 × Z2, having only one unit, cannot be generated by its units. It turns out, in the general theory of rings, that this is essentially the only example. In this note, we give an elementary proof of “A finite commutative ring with nonzero identity is generated by its units if and only if it cannot have Z2 ×Z2 as a quotient.” The proof uses graph theory, and offers, as a byproduct, that in this case, every element is the sum of at most three units.

Let φ: (R, 𝔪) → (S, 𝔫) be a flat local homomorphism of rings. In this paper, we prove: (1) If dim S/𝔪S > 0, then S is a filter ring if and only if R and k(𝔭) ⊗R𝔭 S𝔮 are Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭= 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. (2) If dim S/𝔪S = 0, then S is a filter ring if and only if R is a filter ring and k(𝔭) ⊗R𝔭 S𝔮 is Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭 = 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. As an application, it is shown that for a k-algebra R and an algebraic field extension K of k, if K ⊗k R is locally equidimensional, then R is a locally filter ring if and only if K ⊗k R is a locally filter ring.

Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. Let $R\Join I$ be the subring of $R\times R$ consisting of the elements $(r,r+i)$ for $r\in R$ and $i\in I$. We study the diameter and girth of the zero-divisor graph of the ring $R\Join I$., Comment: 9 pages

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnected flag complex with a $(t-2)$-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai., Comment: 6 pages

Let R be a commutative Noetherian ring, α an ideal of R and M a finitely generated R-module. Let t be a non-negative integer. It is known that if the local cohomology module is finitely generated for all i < t, then is finitely generated. In this paper it is shown that if is Artinian for all i < t, then need not be Artinian, but it has a finitely generated submodule N such that /N is Artinian.

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a finitely generated R-module of finite Krull dimension n. We describe the (finite) sets [Formula: see text] and [Formula: see text] of primes associated and attached to the highest local cohomology module [Formula: see text] in terms of the local formal behaviour of 𝔞.

This article shows some instances where properties of a local ring are closely connected with the homological properties of a single module.

Let \({\left( {R,\mathfrak{m}} \right)}\) be a commutative Noetherian local ring and let \(\mathfrak{a}\) be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to \(\mathfrak{a}\). As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module \({\text{H}}^{i}_{\mathfrak{a}} {\left( M \right)}\) are finite if one of the following holds: (a) \(R \mathord{\left/ {\vphantom {R \mathfrak{a}}} \right. \kern-\nulldelimiterspace} \mathfrak{a} = 1\), (b) \(\mathfrak{a}\) is a principal ideal.

For a finitely generated module M over a commutative Noetherian local ring (R,𝔪), it is shown that there exist only a finite number of non-isomorphic top local cohomology modules [Formula: see text] for all ideals 𝔞 of R. It is also shown that for a given integer r ≥ 0, if [Formula: see text] is zero for all 𝔭 in Supp (M), then [Formula: see text] for all i ≥ r.

In this article a generalized version of the Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension over a commutative Noetherian ring.

Let $G=W(C_{n})$ be a whiskered cycle graph with edge ideal $I=I(G)$. We prove that for every $s\geq 1$, the equality ${\rm reg}(I^{s})=2s+\lceil \frac{n-1}{2}\rceil-1$ holds.

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length at most d-t then either G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given., Comment: Final version, to appear in J. Pure Appl. Algebra. 9 pages. Uses XY-pic

Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.

Let (R, , k) be a commutative Noetherian local ring with maximal ideal . Recently, Asadollahi and Salarian used relative cohomology functors to define Gorenstein test modules. In this article it is shown that every nonzero finitely generated R-module is a Gorenstein test module.

Assume that ϕ:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ⊗ R S.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $\mathfrak{a}$ be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module ${\text{H}}_\mathfrak{a}^n (M).$

In this paper, we show that the property of being catenary and locally equidimensional descends by flat homomorphism. More precisely, if ϕcolonR → S is a flat homomorphism of Noetherian rings, then S is catenary and equidimensional if R is locally equidimensional, and the rings R/𝔭 ⊗ R S, 𝔭 ∈ Min R, are catenary and locally equidimensional. Let k be a field, A a k-algebra, and K an extension field of k. Then we show that the K ⊗ k A is universally catenary if one of the following holds: A is universally catenary and K a finitely generated extension field of k; A is Noetherian universally catenary and t.d.(K:k)

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal $\fa$ of $R$, two different types of cohomological dimensions of a complex $X$ in a certain subcategory of the derived category, namely $\cd(\fa, X)=\sup\{\cd(\fa, \H_{\ell}(X))-\ell|\ell\in\Bbb Z\}$ and $-\inf{\mathbf R}\G_{\fa}(X)$, where $\cd(\fa, M)=\sup\{\ell\in\Bbb Z|\H^{\ell}_{\fa}(M)\neq 0\}$ for an $R$--module $M$. In this paper, it is shown, among other things, that, for any complex $X$ bounded to the left, $-\inf {\mathbf R}\G_{\fa}(X)\le\cd(\fa, X)$ and equality holds if indeed $\H(X)$ is finitely generated., 13 pages

This note makes a correction to the paper “Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module”.

In this paper we solve a problem, originally raised by Grothendieck, on the properties, i.e. Complete intersection, Gorenstein, Cohen--Macaulay, that are conserved under tensor product of algebras over a field $k$., 6 pages

The purpose of this paper is to investigate some connections between Gorenstein flat and Gorenstein injective dimensions of complexes over different rings.

Let $R$ be a commutative ring. In this paper we study the behavior of Gorenstein homological dimensions of a homologically bounded $R$-complex under special base changes to the rings $R_{x}$ and $R/xR$, where $x$ is a regular element in $R$. Our main results refine some known formulae for the classical homological dimensions. In particular, we provide the Gorenstein counterpart of a criterion for projectivity of finitely generated modules, due to Vasconcelos.

We show that the first non-finitely generated generalized local cohomology modules of a finite modules M and N over a Noetherian ring R with respect to an ideal of R has only finitely many associated primes. This generalizes the corresponding result which is shown in [1] for standard local cohomology module.

We determine a bound for numbers of minimal generators for a class of submodules of a finitely generated module with dimension .

The problem of finding a characterization of Cohen--Macaulay simplicial complexes has been studied intensively by many authors. There are several attempts at this problem available for some special classes of simplicial complexes satisfying some technical conditions. This paper is a survey, with some new results, of some of these developments. The new results about simplicial complexes with Serre's condition are an analogue of the known results for Cohen--Macaulay simplicial complexes.

Let R be a (not necessarily Noetherian) commutative ring and let M be a (not necessarily finitely generated) R-module. We characterize the modules with only finitely many weakly associated primes as those modules M admitting a chain 0 = M0 ⊂ M1 ⊂ ... ⊂ Mn = M of submodules together with prime ideals p1, p2,...,pn such that the set of weakly associated primes of Mi/Mi-1 is equal to {pi} for all 1 ≦ i ≦ n. Let M = gra(M) = ⊕n≧0anM/an+1M be the corresponding graded module over the graded ring R = gra(R) = ⊕n≧0an/an+1. It is shown that the union of the set of weakly associated primes of.....

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext t R (M,R) ≠ 0, see [11]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext t R (M, R) ≠ 0, see [3]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.

We consider classes of ideals which generalize the mixed product ideals introduced by Restuccia and Villarreal, and also generalize the expansion construction by Bayati and the first author \cite{BH}. We compute the minimal graded free resolution of generalized mixed product ideals and show that the regularity of a generalized mixed product ideal coincides with regularity of the monomial ideal by which it is induced.

It is shown that for any Artinian modules M, M∨ is the greatest integer i such that \({\text{H}}_m^i \)(M∨) ≠ 0.

In this paper the concept of the magnitude of modules (dual notion to that of dimension of modules) is introduced, and it is used to define the co-Cohen-Macaulay modules (dual notion to that of Cohen-Macaulay modules)

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$., Comment: 7 pages. To appear in Rocky Mountain Journal of Mathematics

This review paper deals with tensor products of algebras over a field. Let k be a field and A, B be commutative k-algebras. We consider the following question: “Which properties of A and B are conveyed to the k-algebra A⊗ k B?”. This field is still developing and many contexts are yet to be explored. We will restrict the scope of the present survey, mainly, to special rings.

We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition $S_r$. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially $S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially $S_r$ if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first $r$ steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially $S_r$ cycles showing that the only sequentially $S_2$ cycles are odd cycles and, for $r\ge 3$, no cycle is sequentially $S_r$ with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially $S_r$ graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially $S_2$. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially $S_2$. Finally, we propose some questions., Comment: 13 pages