Generalized Factorization in Commutative Rings with Zero-Divisors

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.