Irreducible Divisor Graphs and Factorization Properties of Domains

Abstract

This article examines the connections between the factorization properties of a domain, e.g., unique factorization domain (UFD), finite factorization domain (FFD), and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x): x ∈ D* \ U(D)} when D is an FFD, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x): x ∈ D* \ U(D)} when D is not an FFD. We also introduce an alternate irreducible divisor graph called the compressed irreducible divisor graph and study some of its properties.