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

TL;DR

The paper introduces $n$-OA ideals and $n$-OAF rings as a natural extension of prime and $n$-absorbing notions, linking them to generalised ZPI and OAF rings. It establishes foundational properties, including finiteness of minimal primes over ideals and dimensional restrictions, and proves stability under quotients, localization, and products. A central result is that polynomial and power series extensions force the base ring to be a finite direct product of fields (i.e., a Noetherian von Neumann regular ring), and it characterises the behavior of the ring $A+XB[X]$ under the $n$-OAF framework. The paper further analyses the trivial extension $R=A\propto E$, giving precise conditions under which $R$ is $n$-OAF, including when $A$ is a general ZPI-ring and $E$ is cyclic, as well as a sharp equivalence in the integral-domain case to $A$ being a field. Collectively, these results situate $n$-OAF rings as a robust generalisation of ZPI and OAF rings with broad structural consequences and transfer properties across common ring constructions.

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 $E$.

Theorems & Definitions (52)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof