Infinitely many McCoy rings with non-McCoy subrings

We construct an infinite family of McCoy rings containing subrings that are neither left nor right McCoy.

1. Introduction

A (unital associative) ring \(R\) is called left McCoy (resp. right McCoy) if, whenever \(fg=0\) with \(f,g\in R[x]\) nonzero, there exists a nonzero element \(r\in R\) such that \(rg=0\) (resp. \(fr=0\)). We say that \(R\) is McCoy if \(R\) is both left McCoy and right McCoy.

In 2025, a Math Stack Exchange user posed the question of whether the McCoy property passes to subrings. The purpose of this blogpost is document my negative answer to this question in 2026, which was subsequently added to the Database of Ring Theory.

2. The construction

Let \(k\) be an integral domain. We construct a (unital associative) ring \(R\) as follows. The underlying additive group of \(R\) is \(M_2(k)\), the group of \(2\times 2\) matrices with entries in \(k\). We equip \(R\) with the ring multiplication \[AB:=\begin{pmatrix} a_{11}b_{11} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} &a_{22}b_{22} \end{pmatrix}.\] Often, \(R\) is called a Morita context ring or generalized matrix ring whose associated bimodule maps are both zero. Also, if \(k=\mathbb F_2\) is the finite field of order \(2\), then \(R\) is isomorphic to this ring.

In the following, let \(E_{11}\), \(E_{12}\), \(E_{21}\), and \(E_{22}\) denote the usual elementary matrices in \(R\).

Theorem. The ring \(R\) is McCoy, but \(R\) contains a subring \(S\) that is neither left McCoy nor right McCoy.

Proof. Let \(S:= T_2(k)\) be the subset of upper triangular matrices in \(R\), which is a subring with its usual multiplication. We claim that \(S\) is neither left McCoy nor right McCoy. Indeed, consider the polynomials \[f(x):= E_{12}x+E_{11},\qquad g(x):= -E_{12}x+E_{22}\] in \(S[x]\). Evidently, \(fg=0\). Since \(r\) is upper triangular, it is straightforward to show that if \(r\in S\) and \(rg=0\) or \(fr=0\), then \(r=0\), as desired. That is, \(S\) is neither left McCoy nor right McCoy.

Now, we prove that \(R\) is McCoy. Let \(f,g\in R[x]\cong M_2(k[x])\) be nonzero elements, considered as \(2\times 2\) matrices with polynomial entries \(f_{ij},g_{ij}\in k[x]\). Then the equation \(fg=0\) means that the polynomials \[f_{11}g_{11},\quad \varphi:= f_{11}g_{12}+f_{12}g_{22},\quad \psi:= f_{21}g_{11}+f_{22}g_{21},\quad \text{and}\quad f_{22}g_{22}\] in \(k[x]\) all vanish. Since \(k[x]\) is an integral domain, we must have \(f_{11}=0\) or \(g_{11}=0\), and \(f_{22}=0\) or \(g_{22}=0\).

We claim that \(R\) is right McCoy. If \(f_{11}\neq 0\) and \(f_{22}\neq 0\), then \(g_{11}=0\) and \(g_{22}=0\), so the equations \(\varphi=0\) and \(\psi=0\) imply that \(g_{12}=0\) and \(g_{21}=0\) as well. That is, \(g=0\), which is a contradiction. Therefore, \(f_{11}\) and \(f_{22}\) cannot both be nonzero. If \(f_{11}=0\), then a quick calculation shows that \(fE_{12}=0\); if \(f_{22}=0\), then \(fE_{21}=0\). Hence, \(R\) is right McCoy.

Showing that \(R\) is left McCoy is similar. Here are the details for the sake of convenience: if \(g_{11}\neq 0\) and \(g_{22}\neq 0\), then \(f_{11}= 0\) and \(f_{22}= 0\), so \(f_{12}=0\) and \(f_{21}=0\) as well. That is, \(f=0\), which is a contradiction. If \(g_{11}=0\), then \(E_{21}g=0\); if \(g_{22}=0\), then \(E_{12}g=0\). Hence, \(R\) is left McCoy. QED.