Related papers: On $1$-absorbing $\delta$-primary ideals

For the approx.ideal W of the approx.commutative ring R with unity in a descriptive relator space, after introducing the approx. prime ideal in [], this work demonstrates some special properties of the approx.ideals-specifically, the…

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal…

Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal…

Let $R$ be a commutative ring with identity. An ideal $I$ of $R$ is said to be a big ideal (resp. an upper big ideal) if whenever $J\subsetneqq I$ (resp. $I\subsetneqq J$), $J^{n}\subsetneqq I^{n}$ (resp. $I^{n}\subsetneqq J^{n}$) for every…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all…

Let $R$ be a commutative ring with. The purpose of this paper is to introduce and investigate cubes-difference factor absorbing ideals of R as a generalization of prime ideals.

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…

Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then…

In this paper, we define the concept $I-$prime hyperideal in a multiplicative hyperring $R$. A proper hyperideal $P$ of $R$ is an $I-$prime hyperideal if for $a, b \in R$ with $ab \subseteq P-IP$ implies $a \in P$ or $b \in P$. We provide…

Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S$-$n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an…

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…

In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $\phi$-$\delta$-$r$-hyperideal. Let $\Re$ be a commutative Krasner hyperring with nonzero identity. Given an expansion $\delta$ of…

Let R be a multiplicative hyperring. In this paper, we define the concept of 1-absorbing prime hyperideals which is a generalization of the prime hyperideals. Several properties of the hyperideals are provided. Moreover, we introduce the…

In this note we show that in a commutative ring $R$ with unity, for any $n > 0$, if $I$ is an $n$-absorbing ideal of $R$, then $(\sqrt{I})^{n} \subseteq I$.

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…

An ideal I of a commutative ring R is said to be irreducible if it cannot be written as the intersection of two larger ideals. A proper ideal I of a ring R is said to be strongly irreducible if for each ideals J, K of R, J\cap K\subseteq I…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…

In this article, we show that Mori domains, pseudo-valuation domains, and $n$-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain $R$ is a…

In this study, we introduce the concept of "uniformly 2-absorbing primary ideals" of commutative rings, which imposes a certain boundedness condition on the usual notion of 2-absorbing primary ideals of commutative rings. Then we…