arXiv subjects

Shokoufeh Habibi

Publications and source records attributed to Shokoufeh Habibi.

5 recordsLinked to original sources

arXiv2020

Domination number in the annihilating-submodule graph of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(τ_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.

math.AC↗

arXiv2020

The Zariski topology-graph of modules over commutative rings II

Let $M$ be a module over a commutative ring $R$. In this paper, we continue our study about the Zariski topology-graph $G(τ_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). For a non-empty subset $T$ of $Spec(M)$, we obtain useful characterizations for those modules $M$ for which $G(τ_T)$ is a bipartite graph. Also, we prove that if $G(τ_T)$ is a tree, then $G(τ_T)$ is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $χ(G(τ_T))$ and $ω(G(τ_T))$.

math.AC↗

arXiv2016

The annihilating-submodule graph of modules over commutative rings II

Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We prove that if AG(M) is a tree, then either AG(M) is a star graph or a path of order 4 and in the latter case M\cong F\times ?S, where F is a simple module and S is a module with a unique non-trivial submodule. Moreover, we prove that if M is a cyclic module with at least three minimal prime submodules, then gr(AG(M)) = 3 and for every cyclic module M, cl(AG(M))\geq |Min(M)|.

math.AC↗

arXiv2016

The annihilating-submodule graph of modules over commutative rings

Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We obtain useful characterizations for those modules M for which either AG(M) is a complete (or star) graph or every vertex of AG(M) is a prime (or maximal) submodule of M. Moreover, we study coloring of annihilating-submodule graphs.

math.AC↗

arXiv2015

The Zariski topology-graph on the maximal spectrum of modules over commutative rings

Let M be a module over a commutative ring and let Spec(M) (resp. Max(M)) be the collection of all prime (resp. maximal) submodules of M. We topologize Spec(M) with Zariski topology, which is analogous to that for Spec(R), and consider Max(M) as the induced subspace topology. For any non-empty subset T of Max(M), we introduce a new graph G(τ^{m}_{T})called the Zariski topology-graph on the maximal spectrum of M. This graph helps us to study the algebraic (resp. topological) properties of M (resp. Max(M)) by using the graph theoretical tools.

math.AC↗