Commutative rings with one-absorbing factorization

TL;DR

This paper investigates commutative rings where every proper ideal factors into 1-absorbing prime ideals, revealing that such rings have dimension at most one and specific properties in local domains.

Contribution

It introduces the class of OAF-rings where all proper ideals factor into 1-absorbing prime ideals, expanding the understanding of ideal factorizations in commutative rings.

Findings

OAF-rings have dimension at most one

Local OAF-domains are atomic with universal square maximal ideals

Characterization of rings where every ideal factors into 1-absorbing prime ideals

Abstract

Let $R$ be a commutative ring with nonzero identity. A. Yassine et al. defined in the paper (Yassine, Nikmehr and Nikandish, 2020), the concept of $1$-absorbing prime ideals as follows: a proper ideal $I$ of $R$ is said to be a $1$-absorbing prime ideal if whenever $xyz\in I$ for some nonunit elements $x,y,z\in R$, then either $xy\in I$ or $z\in\ I$. We use the concept of $1$-absorbing prime ideals to study those commutative rings in which every proper ideal is a product of $1$-absorbing prime ideals (we call them $OAF$-rings). Any $OAF$-ring has dimension at most one and local $OAF$-domains $(D,M)$ are atomic such that $M^2$ is universal.