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Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))

Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.

Let $R$ be a commutative ring with nonzero identity and let $M$ be a unitary $R$-module. The essential graph of $M$, denoted by $EG(M)$ is a simple undirected graph whose vertex set is $Z(M)\setminus {\rm Ann}_R(M)$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm Ann}_{M}(xy)$ is an essential submodule of $M$. Let $r({\rm Ann}_R(M))\not={\rm Ann}_R(M)$. It is shown that $EG(M)$ is a connected graph with ${\rm diam}(EG(M))\leq 2$. Whenever $M$ is Noetherian, it is shown that $EG(M)$ is a complete graph if and only if either $Z(M)=r({\rm Ann}_R(M))$ or $EG(M)=K_{2}$ and ${\rm diam}(EG(M))= 2$ if and only if there are $x, y\in Z(M)\setminus {\rm Ann}_R(M)$ and $\frak p\in{\rm Ass}_R(M)$ such that $xy\not \in \frak p$. Moreover, it is proved that ${\rm gr}(EG(M))\in \{3, \infty\}$. Furthermore, for a Noetherian module $M$ with $r({\rm Ann}_R(M))={\rm Ann}_R(M)$ it is proved that $|{\rm Ass}_R(M)|=2$ if and only if $EG(M)$ is a complete bipartite graph that is not a star.

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AGE(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ⊂ ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and N be a finitely generated R-module with Supp RN= V (𝔞) for some [Formula: see text]. It is shown that if [Formula: see text] for all i < t and all j < t-i, then [Formula: see text] for all i < t. Let S be the class of all R-modules N with dim R N ≤ k, where k is an integer. It is proved that if [Formula: see text] for all i < t and all [Formula: see text], then [Formula: see text] for all i < t. It follows that [Formula: see text].

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

The compressed annihilator graph??of a commutative ring

Let R be a Noetherian ring, M a finitely generated R-module and N an arbitrary R-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal I of R, and prove that, for all nonnegative integers r, t and all p ? Spec(R) the Bass number ?r(p,HtI (M,N)) is bounded above by ?tj=0?r(p, t?jExtR (M,HjI (N))). A corollary is that Ass (HtI (M,N)? Utj=0 Ass (t?jExtR (M,HjI(N))). In a slightly different direction, we also present some well known results about generalized local cohomology modules.

Let R be a Noetherian ring and I, J two ideals of R. Let M be a ZD-module such that Hom R (R/I, M/L) is finitely generated for any submodule L of M. It is shown that if t is an integer such that is minimax for all i 0, then is not finitely generated.

In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:RM. Let N be a 2-absorbing submodule of M. It is shown that N:RM is a 2-absorbing ideal of R and either Ass R(M/N) is a totally ordered set or Ass R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass R(M/N) a totally ordered set, then N is 2-absorbing whenever N:RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.

Let R be a commutative ring, I a finitely generated ideal of R, and M a zero-divisor R-module. It is shown that the M-grade of I defined by the Koszul complex is consistent with the definition of M-grade of I defined by the length of maximal M-sequences in I. Also, it is shown that, if R is Noetherian, then for any submodule N of M, the following conditions are equivalent: (1) is finite for all finitely generated submodules L of ; (2) If are finitely generated, then is \p0000,finite for all finitely generated submodules L of .

The purpose of this paper is to establish connection between the minimal flat resolutions of Artinian modules and the concept of cosequences in commutative algebra. The main result of this paper leads to a characterization of local Cohen–Macaulay rings in terms of vanishing property of the dual Bass numbers of certain Artinian modules.

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form , where is the completion of a free -module. Also, we construct a minimal flat resolution for R/xR-module 0: from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = . This result leads to a description of the structure of a minimal flat resolution for , nth local cohomology module of R with respect to the ideal , over a local Cohen-Macaulay ring (R, ) of dimension n.

Let R be a commutative Noetherian ring, I an ideal of R, M be a finitely generated R-module and t be a non-negative integer. In this paper, we introduce the concept of I, M-minimax R-modules. We show that $ \text{ Hom}_R(R/I,\, H^t_I(M,\, N)/K) $ is I,M-minimax, for all I,M-minimax submodules K of $ H^t_I(M,\, N) $, whenever N and $ H_{I}^{0}(M) $, $ H_{I}^{1}(M), \, \cdots , \, H_{I}^{t-1}(M) $ are I, M-minimax R-modules. As consequence, it is shown that $ \text{ Ass}_R H^t_I(M,\, N)/K $ is a finite set.