On finite molecularization domains

Abstract

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 $k\left[X^2,X^3\right]$, 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 $R\left[X\right]$ is an FMD.