$\tau$-Complete 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. There are many ways authors have decided to study factorization when zero-divisors present. This paper focuses on the method $\tau$-complete factorizations developed by D.D. Anderson and A. Frazier. There is a natural way to extend $\tau$-complete factorization to commutative rings with zero-divisors. $\tau$-complete factorization is a natural way to think of refining factorizations into smaller pieces until one simply cannot refine any further. We see that this notion translates well into the case of commutative rings with zero-divisors and there is an interesting relationship between the $\tau$-complete finite factorization properties and the original $\tau$-finite factorization properties in rings with zero-divisors developed by the author in 2012.