On Weakly 1-Absorbing Prime Ideals

TL;DR

This paper introduces and explores the properties of weakly 1-absorbing prime ideals in commutative rings, providing characterizations, special cases, and applications to rings of continuous functions.

Contribution

It defines weakly 1-absorbing prime ideals, studies their properties, characterizes rings where all ideals are such, and examines their behavior in rings of continuous functions.

Findings

Characterization of weakly 1-absorbing prime ideals

Identification of rings where all proper ideals are weakly 1-absorbing prime

Analysis of these ideals in rings of continuous functions

Abstract

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each nonunits $x, y, z \in A$ with $0\neq xyz \in P$, then either $xy \in P$ or $z \in P$. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in $C(X)$, which is the ring of continuous functions of a topological space X.