Matlis dual of local cohomology modules

summary:Let $(R,\mathfrak m)$ be a commutative Noetherian local ring, $\mathfrak a$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $\mathfrak a M\neq M$ and ${\rm cd}(\mathfrak a,M) - {\rm grade}(\mathfrak a,M)\leq 1$, where ${\rm cd}(\mathfrak a,M)$ is the cohomological dimension of $M$ with respect to $\mathfrak a$ and ${\rm grade}(\mathfrak a,M)$ is the $M$-grade of $\mathfrak a$. Let $D(-) := {\rm Hom}_R(-,E)$ be the Matlis dual functor, where $E := E(R/\mathfrak m)$ is the injective hull of the residue field $R/\mathfrak m$. We show that there exists the following long exact sequence \begin {eqnarray*} 0 \longrightarrow & H^{n-2}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \longrightarrow D(M) \\ \longrightarrow & H^{n-1}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n+1}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \\ \longrightarrow & H^{n}_{\mathfrak a}(D(H^{n-1}_{(x_1, \ldots ,x_{n-1})}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n-1}_\mathfrak (M))) \longrightarrow \ldots , \end {eqnarray*} where $n:={\rm cd}(\mathfrak a,M)$ is a non-negative integer, $x_1, \ldots ,x_{n-1}$ is a regular sequence in $\mathfrak a$ on $M$ and, for an $R$-module $L$, $H^i_{\mathfrak a}(L)$ is the $i$th local cohomology module of $L$ with respect to $\mathfrak a$

The annihilator ideal graph of a commutative ring

Let $R$ be a commutative ring with nonzero identity and $I$ be a proper ideal of $R$. The annihilator graph of $R$ with respect to $I$, which is denoted by $AG_{I}(R)$, is the undirected graph with vertex-set $V(AG_{I}(R)) = \lbrace x\in R \setminus I : xy \in I\$ for some$\ y \notin I \rbrace$ and two distinct vertices $x$ and $y$ are adjacent if and only if $A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y)$, where $A_{I}(x) = \lbrace r\in R : rx\in I\rbrace$. In this paper, we study some basic properties of $AG_I(R)$, and we characterise when $AG_{I}(R)$ is planar, outerplanar or a ring graph. Also, we study the graph $AG_{I}(\mathbb{Z}_{n})$, where $Z_n$ is the ring of integers modulo $n$

On the matroidal path ideals

We prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of $t$-path ideals of a complete multipartite graph.Comment: This paper has been published in Journal of Algebra and Its Application

Classes of normally and nearly normally torsion-free monomial ideals

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all normally torsion-free $t$-spread principal Borel ideals that can also be viewed as edge ideals of uniform multipartite hypergraphs

On the support of general local cohomology modules and filter regular sequences

Let R be a commutative Noetherian ring with non-zero identity and a an ideal of R. In the present paper, we examine the question whether the support of Hn a (N;M) must be closed in Zariski topology, where Hn a (N;M) is the nth general local cohomology module of nitely generated R-modules M and N with respect to the ideal a

Some results on the annihilator graph of a commutative ring

summary:Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \geq 1$