Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$-ideal of $R$ if for $a,b\in R$, $ab\in I$ and $a\notin \sqrt{0}$ imply $b\in I$. We give a new generalization of the concept of $n$-ideals by defining a proper ideal $I$ of $R$ to be a semi $n$-ideal if whenever $a\in R$ is such that $a^2\in I$, then $a\in \sqrt{0}$ or $a\in I$. We give some examples of semi $n$-ideal and investigate semi $n$-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of...

Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by $\mathrm{AG}\left(R\right)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{ann}}_R\left(xy\right)\ne {\mathrm{ann}}_R\left(x\right)\cup {\mathrm{ann}}_R\left(y\right)$, where for $z\in R$, ${\mathrm{ann}}_R\left(z\right)=\{r\in R:rz=0\}$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator...