On the integral closure of a half-factorial domain
A half-factorial domain (HFD), \(R\), is an atomic integral domain where given any two products of irreducible elements of \(R\): \(\alpha_1 \alpha_2\dots \alpha_n= \beta_1\beta_2 \dots\beta_m\) 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-domain is again a HFD [\textit{S. T. Chapman} and \textit{J. Coykendall} in: Non-Noetherian commutative ring theory, Math. Appl., Dordr. 520, 97-115 (2000; Zbl 0987.13010)]. In this paper we produce an example that demonstrates that the integral closure of a HDF does not even have to be atomic. We investigate the failure of this conjecture closely and highlight some cases where the conjecture does indeed hold.
scientific article; zbMATH DE number 1901793