arxiv: 2604.07465 · v1 · submitted 2026-04-08 · 🧮 math.AC

Recognition: unknown

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

Haotian Ma

Pith reviewed 2026-05-10 17:25 UTC · model grok-4.3

classification 🧮 math.AC

keywords reduced ringintegrally closed ringMcCoy ringlocalizationcommutative ringdirect productzero-divisors

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The pith

There exists a reduced and integrally closed commutative ring whose localizations at all maximal ideals are McCoy rings, but the ring itself is neither McCoy nor locally a domain.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an explicit commutative ring R that is reduced and integrally closed. For every maximal ideal p, the localization R_p is itself an integrally closed McCoy ring, where the McCoy condition requires that any finitely generated ideal of zero-divisors has nonzero annihilator. However, R fails to be a McCoy ring globally and is not a domain after localization at every prime. The construction achieves these mixed local and global behaviors through a direct product of two carefully chosen rings.

Core claim

The central discovery is the existence of a ring R that is reduced and integrally closed, with the property that each localization at a maximal ideal is an integrally closed McCoy ring, while R itself fails to be a McCoy ring and is not locally a domain. The construction uses the direct product of a ring example that has domain localizations together with a finitely generated ideal of zero-divisors with zero annihilator, and a local integrally closed McCoy ring that is not a domain to ensure the local McCoy property holds at all maximal ideals while the global McCoy condition fails.

What carries the argument

The direct product construction combining a ring example with domain localizations and a finitely generated ideal of zero-divisors with zero annihilator, together with a local integrally closed McCoy ring that is not a domain, which preserves the required local McCoy property at maximal ideals.

If this is right

The polynomial ring in one variable over R is integrally closed.
The McCoy property can hold at every maximal localization without holding for the ring itself.
A reduced integrally closed ring need not be locally a domain.
Local McCoy behavior at maximal ideals does not force the ring to be McCoy globally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This construction separates the McCoy condition from being locally a domain in the class of reduced integrally closed rings.
Direct products may be used to isolate other global ring properties from their local counterparts at maximal ideals.
One could test whether the same separation holds when localizing at non-maximal primes instead of only maximal ideals.

Load-bearing premise

The direct product of the example ring with domain localizations and the local McCoy ring preserves the McCoy property at all localizations while the global ring remains non-McCoy.

What would settle it

If a maximal ideal localization in the constructed ring fails to satisfy the McCoy condition, or if the global ring turns out to have the McCoy property, the central claim would be false; this can be checked by testing whether a specific finitely generated ideal of zero-divisors has zero annihilator in R.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper answers an open problem in ring theory with an explicit direct product example that appears to hold up.

read the letter

This paper settles Problem 9 by constructing an explicit reduced integrally closed ring R that is not McCoy and not locally a domain, but has the property that every localization at a maximal ideal is an integrally closed McCoy ring. It does this by taking the direct product of Akiba's Nagata-type example and a specific local integrally closed McCoy ring that is not a domain. The construction is straightforward. Maximal ideals in the product are either of the form p times the second ring or the first ring times q. Localizing at the first type gives a localization of Akiba's ring, which is a domain and thus McCoy. Localizing at the second type gives back the chosen local ring, which is McCoy by assumption. The global non-McCoy behavior comes from the ideal of zero-divisors in the first component, which remains finitely generated with zero annihilator in the product. As a bonus, this implies R[X] is integrally closed. What works well here is the explicitness and the way the direct product separates the local and global behaviors without introducing new complications. The argument relies on known properties of the components rather than inventing new ones, which keeps it grounded. The use of Huckaba's criterion for the polynomial extension is a nice touch that follows naturally. The soft spots are minor. One has to trust that the chosen local ring B satisfies being integrally closed, McCoy, and not a domain, but the paper presumably verifies this. There are no signs of circular reasoning or unfalsifiable claims. The overall logic holds up as described in the stress-test note. This paper is for commutative algebraists working on ring properties like McCoy conditions or integral closure. Someone looking for concrete counterexamples in this niche would get value from it. It does not have broad impact but cleanly resolves the specific open problem. I recommend sending it to peer review. The construction is checkable, and resolving an open question merits referee attention even if the result is specialized.

Referee Report

0 major / 1 minor

Summary. The paper constructs an explicit commutative ring R as the direct product R = A × B, where A is Akiba's Nagata-type example (an integrally closed reduced ring with integrally closed domain localizations but failing to be McCoy) and B is an explicit local integrally closed McCoy ring that is not a domain. The resulting R is reduced and integrally closed; its localizations at maximal ideals (of the form p × B or A × q) are integrally closed McCoy rings (isomorphic to A_p or B); yet R itself is not McCoy (via the finitely generated ideal I × B of zero-divisors with zero annihilator inherited from A) and not locally a domain. This affirmatively answers Problem 9 in Open Problems in Commutative Ring Theory, and implies R[X] is integrally closed by Huckaba's criterion.

Significance. If the verifications hold, the result is significant as it supplies a concrete counterexample separating local and global McCoy behavior in the integrally closed reduced setting, directly resolving an open problem. The direct-product technique for preserving local McCoy properties while retaining a global zero-divisor ideal with zero annihilator is a standard and verifiable method in commutative algebra; the explicitness of the construction (with independent properties of A and B) strengthens its utility as a reference example.

minor comments (1)

The consequence that R[X] is integrally closed via Huckaba's criterion is stated in the abstract and conclusion; a brief recall of the criterion or a precise citation would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of the construction, and the recommendation to accept. The report correctly identifies how the direct product of Akiba's example with the local McCoy ring yields the desired global and local properties while resolving the open problem.

Circularity Check

0 steps flagged

No significant circularity; explicit construction is self-contained

full rationale

The paper constructs R explicitly as the direct product R = A × B, where A is Akiba's independently known Nagata-type example (reduced, integrally closed, with domain localizations and a finitely generated zero-divisor ideal of zero annihilator) and B is an explicitly given local integrally closed McCoy ring that is not a domain. Localizations at maximal ideals of R are isomorphic to A_p (domain, hence McCoy and integrally closed) or to B (McCoy and integrally closed by construction of B). The global failure of the McCoy property is inherited directly from the corresponding ideal in A. No derivation step reduces by definition, fitted parameter, or self-citation chain to the target claim; the verifications rely on standard facts about direct products and localizations together with the independently established properties of A and B.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard axioms of commutative algebra and the properties of direct products; no free parameters or new entities are introduced beyond the explicit rings being combined.

axioms (1)

standard math Commutative rings satisfy the usual axioms of associativity, commutativity, distributivity, and existence of additive and multiplicative identities.
The paper works entirely within the standard framework of commutative ring theory.

pith-pipeline@v0.9.0 · 5458 in / 1254 out tokens · 71098 ms · 2026-05-10T17:25:03.176929+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

Akiba,Integrally-closedness of polynomial rings, Japan

T. Akiba,Integrally-closedness of polynomial rings, Japan. J. Math.6(1980), 67–75

1980
[2]

Cahen, M

P.-J. Cahen, M. Fontana, S. Frisch, and S. Glaz,Open problems in commutative ring theory, in Commutative Algebra: Expository Papers Dedicated to David F. Anderson on His Seventieth Birthday, Springer, New York, 2014, pp. 1–25

2014
[3]

J. A. Huckaba,Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988. 6

1988