Generalized Factorization in Commutative Rings with Zero-Divisors

TL;DR

This paper extends the concept of $ au$-factorization from integral domains to commutative rings with zero-divisors, classifying rings with certain finite factorization properties.

Contribution

It synthesizes existing work on $ au$-factorization and zero-divisors, extending the notion to broader classes of rings and classifying those with finite factorization properties.

Findings

Extended $ au$-factorization to rings with zero-divisors

Classified rings satisfying finite factorization properties

Analyzed specific $ au$ relations in rings with zero-divisors

Abstract

Much work has been done on generalized factorization techniques in integral domains, namely $\tau$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This paper seeks to synthesize work done in these two areas and extend the notion of $\tau$-factorization to commutative rings that need not be domains. In addition, we look into particular types of $\tau$ relations, which are interesting when there are zero-divisors present. We then proceed to classify commutative rings that satisfy the finite factorization properties given in this paper.