Factorization in commutative rings with zero divisors

Abstract

The purpose of this paper is to study factorization in commutative rings with zero divisors with particular emphasis on how the theory of factorization in integral domains is similar to and different from the theory for commutative rings. Various notions of “associate” are considered. Each form of “associate” leads to a type of “irreducible” element, and each type of “irreducible” element leads to a form of atomicity (elements being products of that type of irreducible element) and unique factorization. Numerous examples are given, including an example of an atomic LCM ring which does not satisfy ACCP or have unique factorization. Factorization in polynomial rings and power series rings is considered. © 1996 Rocky Mountain Mathematics Consortium.