  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.
