Comparison of graphs associated to a commutative Artinian ring

Proceedings of the Indian Academy of Sciences. Mathematical sciences

MASOUD GHORAISHI^1¹ 

[1] Department of Mathematics, Bu Ali Sina University, Hamedan, Iran^1

关键词: Zero-divisor graph;    co-maximal graph;    Boolean ring;    Frobenius ring;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let $R$ be a commutative ringwith $1 \neq 0$ and the additive group $R^{+}$. Several graphs on $R$ have been introduced by many authors, among zero-divisor graph $\Gamma_{1}(R)$, co-maximal graph $\Gamma_{2}(R)$, annihilator graph $AG(R)$, total graph $T (\Gamma(R))$, cozero-divisors graph $\Gamma_{c}(R)$, equivalence classes graph $\Gamma_{E}(R)$ and the Cayley graph Cay$(R^{+}, Z^{\ast}(R))$.Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to Cay$(R^{+}, Z^{\ast}(R))$. Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when $R$ is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if $R$ has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results,we prove that for a commutative finite ring $R$ with $|Max(R)| = n \geq 3$, $\Gamma_{1}(R) \simeq \Gamma_{2}(R)$ if and only if $R \simeq \mathbb{Z}^{n}_{2}$; if and only if $\Gamma_{1}(R) \simeq \Gamma_{E}(R)$. Also then annihilator graph is identical to the cozero-divisor graph if and only if $R$ is a Frobenius ring.

【 授权许可】

CC BY