A fundamental theorem of modular arithmetic

Abstract

The Fundamental Theorem of Arithmetic is a statement about the uniqueness of factorization in the ring of integers. The notation and proof easily generalize to uniqueness of factorization in principal ideal domains. Factorization, unique and otherwise, has been well-studied in integral domains. Less well-studied is factorization in rings that contain nonzero zerodivisors. In this article we give a theorem analogous to the Fundamental Theorem of Arithmetic, but for rings of the form D/(n) where D is a principal ideal domain and n is a nonzero nonunit of D.