When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?

详细信息    查看全文

• 作者：S. Visweswaran ; ^{s_visweswaran2006@yahoo.co.in" class="auth_mail" title="E-mail the corresponding author} ; Hiren D. Patel
• 关键词：Annihilating ideal graph of a commutative ring ; Complemented graph ; Zero-dimensional quasisemilocal ring ; Special principal ideal ring
• 刊名：Arab Journal of Mathematical Sciences
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：22
• 期：1
• 页码：1-21
• 全文大小：287 K

文摘

Let R$R$ be a commutative ring with identity. Let A(R)$A\left(R\right)$ denote the collection of all annihilating ideals of R$R$ (that is, A(R)$A\left(R\right)$ is the collection of all ideals I$I$ of R$R$ which admits a nonzero annihilator in R$R$). Let AG(R)$AG\left(R\right)$ denote the annihilating ideal graph of R$R$. In this article, necessary and sufficient conditions are determined in order that AG(R)$AG\left(R\right)$ is complemented under the assumption that R$R$ is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R$R$ such that AG(R)$AG\left(R\right)$ is complemented under the assumption that A(R)$A\left(R\right)$ contains at least two nonzero ideals.