Relative primeness

In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa₁ · · · a_n where λ is in U (D) and a_i is τ-related to a_j, denoted a_i τ a_j, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization.

Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)_v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b)^∗ = D. We then write a τ_max b (resp. a τ_v b, a τ_{[ ]} b, or a τ_∗ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).