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A ring $R$ is said to be right McCoy if whenever $fg=0$ for nonzero $f,g\in R[x]$, there exists $r\ne 0$ in $R$ such that $fr=0$. Left McCoy rings defined similarly. A ring that is both left and right McCoy is called McCoy.

Some ring properties passes to subrings like commutativity, Armendariz etc.

Does the McCoy property also pass to subrings?

In DaRT, it is given that the McCoy property is stable under products and passes to polynomial rings. Is there any example of a subring of a McCoy ring which is not McCoy?

If we try to prove that subrings are also McCoy, here is my approach: Let $S$ be a subring of a McCoy ring $R$. Take $fg=0$ for some nonzero $f,g\in S[x]$. Since $f,g$ also belong to $R[x]$, there exists nonzero $r\in R$ such that $fr=0$. But what is the guarantee that this $r$ also belongs to $S$?

Martin Brandenburg
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Maths Wizard
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  • Just taking a look at the website you linked, it seems to say that $M_n(\mathbb{Q})$ is not McCoy (I guess for $n \geq 2$ ?), but $\mathbb{Q}$ is, so probably that will be an example. – sl_09 Feb 19 '25 at 19:28
  • But if $M_n(\mathbb{Q})$ is not McCoy, how it prove that subrings are not McCoy. $M_n(\mathbb{Q})$ is not a subring of $\mathbb{Q}$ – Maths Wizard Feb 19 '25 at 19:38
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    Ah apologies, I misread you and thought you were asking for the other way around. – sl_09 Feb 19 '25 at 19:41
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    @MartinBrandenburg okay I fixed it. – Maths Wizard Feb 19 '25 at 21:26
  • Counterexamples, if any, should be non-reversible and non-Armendariz (these properties imply McCoy and pass to subrings). – Amateur_Algebraist Feb 20 '25 at 16:39

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