On some embedding of the strong annihilating-ideal graph of commutative rings

Abstract

Let $\mathfrak{S}$ be a commutative ring with unity (CRU) and $W(\mathfrak{S})$ be the set of annihilating-ideals of $\mathfrak{S}$. The strong annihilating-ideal graph of $\mathfrak{S}$, denoted by $SAG(\mathfrak{S})$, is an undirected graph with vertex set $W(\mathfrak{S})^*$. Two vertices $\mathfrak{m}$ and $\mathfrak{n}$ are adjacent if and only if $\mathfrak{m} \cap \text{Ann}(\mathfrak{n}) \neq (0)$ and $\mathfrak{n} \cap \text{Ann}(\mathfrak{m}) \neq (0)$.
In this paper, we first characterize the Artinian commutative rings $\mathfrak{S}$ for which $SAG(\mathfrak{S})$ has outerplanarity index 2. Then, we classify Artinian commutative rings $\mathfrak{S}$ for which $SAG(\mathfrak{S})$ is double toroidal or Klein-bottle. Finally, we determine the book thickness of $SAG(\mathfrak{S})$ for genus at most one.