Commutative rings with $n$-$1$-absorbing prime factorization

TL;DR

This paper introduces the concept of n-OAF rings, a generalization of ZPI and OAF rings, characterizes Noetherian von Neumann regular rings using this concept, and explores its properties in various ring extensions.

Contribution

It defines n-OAF rings and ideals, provides characterizations of specific ring classes, and studies the behavior of the n-OAF property in common ring extensions.

Findings

n-OAF rings generalize ZPI and OAF rings

Characterization of Noetherian von Neumann regular rings via n-OAF property

Analysis of n-OAF property in polynomial, power series, and trivial extension rings

Abstract

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 $a_1a_2\cdots a_n\in I$ or $a_{n+1}\in I$. A commutative ring $R$ is said to be an \textit{$n$-OAF ring} if every proper ideal $I$ of $R$ is a product of finitely many $n$-OA ideals. In fact, $1$-OAF rings and $2$-OAF $2$-OAF-rings are exactly the general ZPI rings and OAF rings, respectively. In addition to giving various properties of $n$-OAF rings, we give a characterization of Noetherian von Neumann regular rings in terms of our new concept. Furthermore, we investigate the $n$-OAF property of some extension of rings such as the polynomial ring $R[X]$, the formal power series ring $R[[X]]$, the ring of $A+XB[X]$, and the trivial extension $R=A\propto E$ of an $A$-module $E$.