On $\phi$-1-Absorbing Prime Ideals

TL;DR

This paper introduces and studies a new class of prime ideals called $\,\phi$-1-absorbing prime ideals in commutative rings, providing properties, characterizations, and conditions for rings where all ideals are of this type.

Contribution

It defines $\,\phi$-1-absorbing prime ideals, explores their properties, and characterizes rings where every proper ideal is of this form.

Findings

Characterization of $\,\phi$-1-absorbing prime ideals.

Properties and properties of rings where all ideals are $\,\phi$-1-absorbing prime.

Conditions under which every proper ideal is $\,\phi$-1-absorbing prime.

Abstract

In this paper, we introduce $\phi$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $\phi:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where $\mathcal{I}(R)$ is the set of all ideals of $R$. A proper ideal $I$ of $R$ is called a $\phi$-1-absorbing prime ideal if for each nonunits $x,y,z\in R$ with $xyz\in I-\phi(I)$, then either $xy\in I$ or $z\in I$. In addition to give many properties and characterizations of $\phi$-1-absorbing prime ideals, we also determine rings in which every proper ideal is $\phi$-1-absorbing prime.