Generalized factorization in $\mathbb{Z}/m\mathbb{Z}$

Abstract

Generalized factorization theory for integral domains was initiated by D. D. Anderson and A. Frazier in 2011 and has received considerable attention in recent years. There has been significant progress made in studying the relation ${\tau }_n$ for the integers in previous undergraduate and graduate research projects. In 2013, the second author extended the general theory of factorization to commutative rings with zero-divisors. In this paper, we consider the same relation ${\tau }_n$ over the modular integers, $\mathbb{Z}∕m\mathbb{Z}$. We are particularly interested in which choices of $m,n\in \mathbb{N}$ yield a ring which satisfies the various ${\tau }_n$-atomicity properties. In certain circumstances, we are able to say more about these ${\tau }_n$-finite factorization properties of $\mathbb{Z}∕m\mathbb{Z}$.