Some results on the strongly annihilating-ideal graph of a commutative ring

Abstract

Let R be a commutative ring with identity and $\mathbb {A}(R)$ be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of R is defined as the graph $\mathrm {SAG}(R)$ with the vertex set $\mathbb {A}(R)^*=\mathbb {A}(R){\setminus }\{0\}$ and two distinct vertices I and J are adjacent if and only if $I\cap \mathrm {Ann}(J)\ne (0)$ and $J\cap \mathrm {Ann}(I)\ne (0)$. We show that if R is a reduced ring with finitely many minimal primes, then $\mathrm {SAG}(R)$ is weakly prefect and an explicit formula for the vertex chromatic number of $\mathrm {SAG}(R)$ is given. Furthermore, strongly annihilating-ideal graphs with finite clique numbers are studied. Finally, we classify that all rings whose $\mathrm {SAG}(R)$ are planar.