Some classes of perfect annihilator-ideal graphs associated with commutative rings

Document Type : Research Paper


1 Department of Mathematics, Roudbar Branch Islamic Azad University, Roudbar, Iran

2 Department of Mathematics, Imam Khomeini International University, Qazvin, Iran


Let $R$ be a commutative ring and let $\Bbb A(R)$ be
the set of all ideals of $R$ with nonzero annihilator.
The annihilator-ideal graph of $R$ is defined as the graph
${A_I}(R)$ with the vertex set $\Bbb A(R)^*=\Bbb A(R)\setminus\{0\}$ and two
distinct vertices $I$ and $J$ are adjacent if and only if
${\rm Ann}_R(IJ)\neq{\rm Ann}_R(I) \cup{\rm Ann}_R(J)$. In this paper, perfectness of
${A_I}(R)$ for some classes of rings is investigated.