Some factorization properties of idealization in commutative rings with zero divisors

TL;DR

This paper investigates the factorization properties of idealization rings in commutative rings with zero divisors, establishing conditions for ACCP, BFR, and UFR properties in these constructions.

Contribution

It provides new characterizations of when idealization rings preserve ACCP, BFR, and UFR properties, extending factorization theory to rings with zero divisors.

Findings

R + M is ACCP iff R is ACCP and M satisfies ACC on cyclic submodules

BF property is not necessarily preserved in idealization

Conditions identified for idealization rings to be BFR and UFR

Abstract

We study some factorization properties of the idealization $R \mathop{(\! + \! )} M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R \mathop{(\! + \! )} M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R \mathop{(\! + \! )} M$ is a BFR. We also characterize the idealization rings which are UFRs.