#### Vol. 9, No. 3, 2016

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

### Austin Mahlum and Christopher Park Mooney

Vol. 9 (2016), No. 3, 379–393
##### 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, $ℤ∕mℤ$. We are particularly interested in which choices of $m,n\in ℕ$ 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 $ℤ∕mℤ$.

##### Keywords
modular integers, generalized factorization, zero-divisors, commutative rings
##### Mathematical Subject Classification 2010
Primary: 13A05, 13E99, 13F15
##### Milestones
Received: 27 September 2014
Revised: 7 April 2015
Accepted: 6 June 2015
Published: 3 June 2016

Communicated by Vadim Ponomarenko
##### Authors
 Austin Mahlum Department of Mathematics Viterbo University La Crosse, WI 54601 United States Christopher Park Mooney Department of Mathematics Westminster College Fulton, MO 65251 United States