On the annihilator graph of group rings
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Let $R$ be a commutative ring with nonzero identity and $G$ be a nontrivial finite group. Also, let $Z(R)$ be the set of zero-divisors of $R$ and, for $a\in Z(R)$, let $\T{ann}(a) = \lbrace r\in R\ \vert \ ra=0\rbrace$. The annihilator graph of the group ring $RG$ is defined as the graph $AG(RG)$, whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices $x$ and $y$ are adjacent if and only if $\T{ann}(xy)\neq \T{ann}(x) \cup \T{ann}(y)$. In this paper, we study the annihilator graph associated to a group ring $RG$.
Abstract
Let $R$ be a commutative ring with nonzero identity and $G$ be a nontrivial finite group. Also, let $Z(R)$ be the set of zero-divisors of $R$ and, for $a\in Z(R)$, let $\T{ann}(a) = \lbrace r\in R\ \vert \ ra=0\rbrace$. The annihilator graph of the group ring $RG$ is defined as the graph $AG(RG)$, whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices $x$ and $y$ are adjacent if and only if $\T{ann}(xy)\neq \T{ann}(x) \cup \T{ann}(y)$. In this paper, we study the annihilator graph associated to a group ring $RG$.
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