An Integrally Closed Reduced Ring with McCoy Localizations That Is Neither McCoy nor Locally a Domain

Abstract

We construct an explicit commutative ring $R$ that is reduced and integrally closed, such that $R_{\mathfrak p}$ is an integrally closed McCoy ring for every maximal ideal $\mathfrak p$ of $R$, while $R$ itself is not a McCoy ring and is not locally a domain. This gives an affirmative answer to Problem~9 in \emph{Open Problems in Commutative Ring Theory}. The construction combines Akiba's Nagata-type example, which already yields an integrally closed reduced ring with integrally closed domain localizations and a finitely generated ideal of zero-divisors with zero annihilator, with an explicit local integrally closed McCoy ring that is not a domain. Taking the direct product of these two rings preserves the required local McCoy property while retaining the global failure of the McCoy condition. As a consequence, $R[X]$ is integrally closed by Huckaba's criterion.The proof presented in this note was completed by Rethlas \cite{Rethlas2604}, a natural-language automated reasoning system; the author was responsible for reviewing and checking the argument.