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...

If $R$ is a commutative ring with identity and $\le$ is defined by letting $a\le b$ mean $ab=a$ or $a=b$, then $(R,\le )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\le )$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_n$ of integers mod $n$ for $n\ge 2$. In particular, if $R$ is reduced, then $(R,\le )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\le )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_3,Z_4$, $Z_2\left[x\right]/x^2{Z}_2\left[x\right]$, or a four element field.

It is well known that to every Boolean ring $ℛ$ can be assigned a Boolean algebra $ℬ$ whose operations are term operations of $ℛ$. Then a symmetric difference of $ℬ$ together with the meet operation recover the original ring operations of $ℛ$. The aim of this paper is to show for what a ring $ℛ$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached...

In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also examine the preservation of diameter of zero-divisor graph under content extensions.