On the integral closure of a half-factorial domain

Abstract

A half-factorial domain (HFD), R, is an atomic integral domain where given any two products of irreducible elements of R: α 1α 2⋯α n=β 1β 2⋯β m then n= m. As a natural generalization of unique factorization domains (UFD), one wishes to investigate which “good” properties of UFDs that HFDs possess. In particular, it has been conjectured that the integral closure of a half-factorial domain is again a HFD (see Non-Noetherian Commutative Ring Theory, Mathematics and its applications, Vol. 520, Kluwer, Dordrecht, 2000, pp. 97–115. for example). In this paper we produce an example that demonstrates that the integral closure of a HFD does not even have to be atomic. We shall investigate the failure of this conjecture closely and highlight some cases where the conjecture does indeed hold.