TY  - JOUR
ID  - Pejman2019
LB  - Pejman2019
AU  - Pejman, S.B.
AU  - Payrovi, Sh.
AU  - Babaei, S.
TI  - The intersection graph of annihilator submodules of a module
ST  - The intersection graph of annihilator submodules of a module
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2019
VL  - 39
IS  - 4
SP  - 577
EP  - 588
DO  - https://doi.org/10.7494/OpMath.2019.39.4.577
UR  - https://doi.org/10.7494/OpMath.2019.39.4.577
AB  - 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 \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).
KW  - prime submodule
KW  - annihilator submodule
KW  - intersection annihilator graph
SN  - 1232-9274
ER  -