On finite molecularization domains

TL;DR

This paper introduces finite molecularization domains (FMDs), an ideal-theoretic analogue of finite factorization domains, characterizing their properties and connections with other factorization concepts, and providing examples and conditions for their occurrence.

Contribution

It defines and characterizes FMDs, explores their relationship with FFDs and FSDs, and identifies conditions under which certain classical rings are FMDs.

Findings

FMDs are characterized by finitely many nonfactorable ideals dividing any nonzero ideal.

The paper connects FMDs with FFDs and FSDs at the local level.

Examples of FMDs include polynomial rings over fields and certain classical constructions.

Abstract

In this paper, we advance an ideal-theoretic analogue of a "finite factorization domain" (FFD), giving such a domain the moniker "finite molecularization domain" (FMD). We characterize FMD's as those factorable domains (termed "molecular domains" in the paper) for which every nonzero ideal is divisible by only finitely many nonfactorable ideals (termed "molecules" in the paper) and the monoid of nonzero ideals of the domain is unit-cancellative, in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop a number of connections, particularly at the local level, amongst the concepts of "FMD", "FFD", and the "finite superideal domains" (FSD's) of Hetzel and Lawson. Characterizations of when $k[X^2, X^3]$, where $k$ is a field, and the classical $D+M$ construction are FMD's are provided. We also demonstrate that if $R$ is a Dedekind domain with the finite norm property, then $R[X]$ is an FMD.