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

This paper constructs an explicit commutative ring that is reduced and integrally closed with locally integrally closed McCoy properties, yet fails to be McCoy or locally a domain globally, thereby affirmatively resolving Problem 9 in *Open Problems in Commutative Ring Theory*.

The Gist

Imagine you are an architect trying to build a very specific type of house. You have a set of strict rules (mathematical properties) you need to follow, but you also need to break a few other rules to prove a point.

This paper is about a mathematician named Haotian Ma who successfully built a "mathematical house" (a ring) that solves a long-standing puzzle. Here is the story of how he did it, using simple analogies.

The Goal: The "Perfect" House with a Flaw

In the world of algebra, there is a concept called a McCoy Ring. Think of this as a rule about how "messy" parts of a house interact.

• The Rule: If you have a small group of "broken" items (zero-divisors) in a room, there must be at least one other item in the house that can "cancel them out" (an annihilator).
• The Puzzle: Mathematicians wondered: Can we build a house that is perfectly stable and well-organized (Integrally Closed and Reduced) everywhere, such that every single room (localization) follows the McCoy rule, yet the house as a whole breaks the McCoy rule?

For a long time, no one knew if such a house could exist. This paper says: Yes, it can.

The Ingredients: Two Different Types of Rooms

To build this house, Ma didn't try to design one giant, complex room. Instead, he built a house out of two distinct wings (a direct product) and glued them together.

Wing A: The "Perfectly Organized" Wing (Akiba's Example)

• What it is: This wing is made of many tiny, perfect, separate rooms (domains).
• The Good News: If you walk into any single room here, it is perfectly stable. It follows all the rules. It is "locally a domain" (a place with no broken items).
• The Bad News: Even though every individual room is perfect, there is a hidden structural flaw in the entire wing. There is a specific collection of broken items that, when you look at the whole wing, have no one to cancel them out.
• Analogy: Imagine a library where every single book is in perfect condition. However, if you try to shelve a specific set of books together, the shelf collapses because of a hidden design flaw in the library's foundation. Locally, everything is fine; globally, it's broken.

Wing B: The "Messy but Friendly" Wing (The Local McCoy Ring)

• What it is: This is a single, small room that is inherently "messy" (it has broken items everywhere), but it follows the McCoy rule perfectly.
• The Good News: In this room, every time you find a broken item, there is always a "fixer" item nearby that cancels it out. It is a "McCoy" room.
• The Bad News: It is not a "perfect" room. It has broken items, so it's not a "domain."
• Analogy: Imagine a workshop full of broken tools. But, for every broken hammer, there is a specific wrench that fixes it. The workshop is chaotic (not a domain), but it is very organized in how it handles the chaos (it is McCoy).

The Construction: Gluing Them Together

Ma took Wing A and Wing B and glued them together to make the final house, Ring R.

Here is why this combination works like magic:

1. The "Local" Check (Looking at one room at a time):

   • If you visit a room in Wing A, it's a perfect domain. Perfect domains are always McCoy. So, these rooms pass the test.
   • If you visit the room in Wing B, it's the "Messy but Friendly" workshop. We already proved it follows the McCoy rule.
   • Result: Every single room in the house follows the McCoy rule. The house is "locally McCoy."

2. The "Global" Check (Looking at the whole house):

   • Now, look at the whole house. The broken items from Wing A (the ones that had no fixers) are still there.
   • The "fixers" from Wing B cannot help Wing A because they are in a different wing. They are like two separate apartments; the plumber in Apartment B can't fix the leak in Apartment A.
   • Result: The house as a whole has a group of broken items with no fixers. Therefore, the whole house is not a McCoy ring.

3. The "Domain" Check:

   • Is the whole house a "perfect domain" (no broken items anywhere)? No, because Wing B is full of broken items. So, the house is "not locally a domain."

The Big Picture

By combining a "perfect but globally broken" structure with a "messy but locally perfect" structure, Ma created a mathematical object that:

• Is stable and well-ordered (Integrally Closed).
• Has no "ghosts" (Reduced).
• Follows the rules in every single room (Locally McCoy).
• But breaks the rules when you look at the whole building (Not McCoy).

Why Does This Matter?

Before this paper, mathematicians had a "Problem 9" asking if such a thing was possible. They suspected it might be, but they couldn't prove it.

This paper is the "blueprint" that proves it exists. It shows that mathematical properties don't always behave the way we expect when you zoom out from a single room to the whole building. Just because every room is perfect doesn't mean the whole building is perfect, and just because the whole building is messy doesn't mean the rooms are messy.

In short: Haotian Ma built a mathematical house that is locally perfect but globally flawed, solving a mystery that had been open for years.

Technical Summary

1. Problem Statement

The paper addresses Problem 9 from Open Problems in Commutative Ring Theory. The problem asks for the existence of a commutative ring $R$ satisfying the following specific criteria:

1. Global Properties: $R$ must be reduced (no non-zero nilpotent elements) and integrally closed.
2. Local Properties: For every maximal ideal $\mathfrak{p}$ of $R$, the localization $R_{\mathfrak{p}}$ must be an integrally closed McCoy ring.
3. Global Failures: Despite the local properties, $R$ itself must not be a McCoy ring, and $R$ must not be locally a domain (i.e., there exists at least one maximal ideal $\mathfrak{p}$ such that $R_{\mathfrak{p}}$ is not a domain).

Context:

• A ring $R$ is a McCoy ring if every finitely generated ideal $I \subseteq Z(R)$ (where $Z(R)$ is the set of zero-divisors) has a non-zero annihilator.
• Previous results by Akiba and Huckaba established that if $R$ is reduced and locally an integrally closed McCoy ring, then the polynomial ring $R[X]$ is integrally closed. However, it was unknown whether the global McCoy condition is necessary for this implication or if a counterexample exists where the local conditions hold but the global condition fails.

2. Methodology

The author constructs the counterexample $R$ using a direct product of two distinct rings, $A$ and $B$, i.e., $R = A \times B$. The construction relies on three main components:

A. The "Akiba Factor" ($A$)

The author utilizes a specific example from Akiba's work (a Nagata-type construction).

• Construction: Let $k$ be a field and $P$ be a set of representatives of irreducible polynomials in $k[X, Y]$. The ring $A$ is constructed as a subring of an infinite product of derived normal rings of quotients of $k[X, Y]$. Specifically, $A = R_{A,0}[u, v]$, where $R_{A,0}$ involves a direct sum of these normalizations.
• Properties of $A$:
  • $A$ is reduced and integrally closed.
  • Every localization $A_{\mathfrak{m}}$ at a maximal ideal is an integrally closed domain.
  • Crucially, $A$ contains a finitely generated ideal $I_A = (u, v)$ contained in $Z(A)$ such that $\text{Ann}_A(I_A) = 0$. Thus, $A$ is not a McCoy ring.

B. The "Local McCoy Factor" ($B$)

The author constructs a local ring $B$ that is integrally closed and McCoy but not a domain.

• Construction: Let $S_i = k[[t]]$ (formal power series) and $m_i = t k[[t]]$. Define $B = k + \bigoplus_{i \ge 1} m_i$ as a subring of the product $\prod S_i$.
• Properties of $B$:
  • $B$ is a local ring with maximal ideal $\mathfrak{m}_B = \bigoplus m_i$.
  • Every element in $\mathfrak{m}_B$ is a zero-divisor (due to finite support in the direct sum), so $Z(B) = \mathfrak{m}_B$.
  • Every non-zero-divisor is a unit, making $B$ integrally closed in its total quotient ring ($Q(B)=B$).
  • $B$ satisfies the McCoy condition: any finitely generated ideal in $\mathfrak{m}_B$ has a non-zero annihilator (constructed by choosing an index outside the finite support of the generators).
  • $B$ is not a domain.

C. The Direct Product ($R = A \times B$)

The final ring is defined as $R = A \times B$. The author verifies that the direct product preserves the necessary properties while combining the failures of the components.

3. Key Contributions and Results

The Main Theorem (Theorem 4)

The paper proves the existence of a ring $R = A \times B$ with the following properties:

1. Reduced and Integrally Closed: Since $A$ and $B$ are reduced and integrally closed, their product $R$ inherits these properties.
2. Local McCoy Property: For any maximal ideal $\mathfrak{p}$ of $R$:
   • If $\mathfrak{p} = \mathfrak{m} \times B$ (where $\mathfrak{m} \in \text{Max}(A)$), then $R_{\mathfrak{p}} \cong A_{\mathfrak{m}}$, which is an integrally closed domain (and thus McCoy).
   • If $\mathfrak{p} = A \times \mathfrak{m}_B$, then $R_{\mathfrak{p}} \cong B$, which is an integrally closed McCoy ring.
   • Result: Every maximal localization is an integrally closed McCoy ring.
3. Not Locally a Domain: At the maximal ideal $\mathfrak{p}_0 = A \times \mathfrak{m}_B$, the localization is $B$, which is not a domain.
4. Not a McCoy Ring: The ideal $J = I_A \times B$ is finitely generated and contained in $Z(R)$. However, $\text{Ann}_R(J) = 0$ because $\text{Ann}_A(I_A) = 0$. Thus, $R$ fails the McCoy condition globally.

Corollary 5

As a consequence of the main theorem and Huckaba's observation, the paper concludes that there exists an integrally closed reduced ring $R$ such that $R[X]$ is integrally closed, even though $R$ is neither a McCoy ring nor locally a domain.

4. Significance

• Resolution of an Open Problem: The paper provides an affirmative answer to Problem 9 in Open Problems in Commutative Ring Theory, settling a long-standing question regarding the relationship between local McCoy properties and global McCoy properties.
• Refinement of Polynomial Ring Theory: It demonstrates that the condition "locally a McCoy ring" is sufficient to ensure that $R[X]$ is integrally closed, even if the ring $R$ itself is not McCoy and not locally a domain. This clarifies the boundaries of Huckaba's criterion.
• Structural Insight: The construction highlights the utility of direct products in commutative algebra for creating rings with mixed local/global behaviors. It shows that the McCoy property is not a "local-to-global" property in the same way that being a domain or being integrally closed can be.

In summary, Haotian Ma successfully constructs a counterexample that decouples the global McCoy property from local McCoy properties and local domain properties, thereby expanding the understanding of integrally closed polynomial rings.