τ-U-factorizations and their graphs in commutative rings with zero divisors

Let R be a commutative ring with identity, and let τ be a relation on the nonzero, non-unit elements of R. In this paper we generalize the definitions of a factorization and a U-factorization via a relation τ and construct a variety of graphs based on these generalizations. These graphs are then examined in an effort to determine ring-theoretic properties.