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

TL;DR

The paper constructs a specific reduced, integrally closed ring with McCoy localizations that is neither McCoy nor locally a domain, answering a key open problem in commutative ring theory.

Contribution

It provides an explicit example of such a ring, combining known constructions to resolve an open problem about McCoy rings and local properties.

Findings

Constructed a reduced, integrally closed ring with McCoy localizations that is not McCoy.

Showed that the ring's polynomial extension is also integrally closed.

Demonstrated the ring's properties through explicit construction and combination of known examples.

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.