Finite Conductor Rings with Zero Divisors

Abstract

The finite conductor property of a domain R— that is the finite generation of the conductor ideals (I: J) for principal ideals I and J of R, came into prominence with the publication of McAdam’s work [35]. The definition of a finite conductor domain appears in an early unpublished version of McAdam’s manuscript, but it appears in print for the first time in [11]. The notion embodies, in its various aspects, both factoriality properties and finiteness conditions. Indeed, the class of domains where (I: J) is always principal for any two principal ideals I and J, is precisely that of Greatest Common Divisor (GCD) domains, while the requirement that each such (I: J) be finitely generated is a necessary condition for the coherence of a domain. For that reason the finite conductor property makes frequent, explicit or implicit, appearance in the literature in two kinds of, occasionally intermingling, investigations: those involving factoriality and those concerned with finiteness, coherent-like conditions, of domains. Regarding investigations involving factoriality: GCD domains were investigated in their own right for a variety of structural properties, as an aspect of properties of various ring constructions, and as a source of generalizations to other classes of rings. Articles [2, 3, 5, 6, 9, 10, 11, 12, 15, 18, 31, 32, 44, 50] provide just a partial list of references on the subject.