Associates, irreducibility, and factorization length in monoid rings with zero divisors

Abstract

In the context of factorization in monoid rings with zero divisors, we study associate relations and the resulting notions of irreducibility and factorization length. Building upon these facts, we determine necessary and sufficient conditions for broad classes of monoid rings to satisfy the ascending chain condition on principal ideals or be a bounded factorization ring. Along the way, we completely characterize the monoid rings that are présimplifiable or domainlike, building upon the results of Ghanem/Wyn-Jones/Al-Ezeh and Anderson/Al-Mallah about group rings with these properties.

Keywords:

• Ascending chain condition on principal ideals
• bounded factorization ring
• monoid ring; irreducible elements
• présimplifiable/domainlike ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F15
• 13A05
• 20M25
• 13P05

Acknowledgments

We would like to sincerely thank the anonymous referees for their careful reading of our paper and their thoughtful suggestions for improvement.