On factorable rings

Abstract

In this short note, we introduce the notions of ``factorable ring" and ``fully factorable ring" for commutative rings based upon the notion of ``factorable domain" advanced by Anderson, Kim and Park~\cite {AKP}. Using a novel sufficient condition for an ideal to be a product of nonfactorable ideals, we classify the Artinian rings that are (fully) factorable. We also explore the intersection of the class of factorable rings with the class of Noetherian rings. An analogue for multiplication rings of a characterization result due to Butts~\cite {HSB} concerning when such a unique factorization occurs is provided.