# Relation between Abelian rings and McCoy rings

First, some basic definitions:

Abelian ring : A ring $$R$$ is said to be abelian if its every idempotent is central.

Armendariz ring : A ring $$R$$ is said to be Armendariz if whenever $$f(x),g(x)\in R[x]$$ satisfy $$f(x)g(x)=0$$ then $$a_ib_j=0,\forall 0\leq i \leq n, 0\leq j \leq m$$, where $$f(x)=\sum_{i=0}^{i=n}a_ix^i$$ and $$g(x)=\sum_{j=0}^{j=m}b_jx^j$$.

McCoy ring: A ring $$R$$ is said to be McCoy if whenever $$f(x),g(x)\in R[x]$$ satisfy $$f(x)g(x)=0$$ then $$a_ir=0$$ and $$sb_j=0,\forall 0\leq i \leq n, 0\leq j \leq m$$, and for some nonzero $$r,s\in R$$ where $$f(x)=\sum_{i=0}^{i=n}a_ix^i$$ and $$g(x)=\sum_{j=0}^{j=m}b_jx^j$$.

Now we know that all Armendariz rings are abelian and also all Armendariz rings are McCoy. So what is the relation between Abelian and McCoy?

## 1 Answer

Camillo, Victor, and Pace P. Nielsen. "McCoy rings and zero-divisors." Journal of Pure and Applied Algebra 212.3 (2008): 599-615.

Theorem 7.1. There exists a McCoy ring with 1 which is not Abelian.

Nielsen, Pace P. "Semi-commutativity and the McCoy condition." Journal of Algebra 298.1 (2006): 134-141.

Section 3 p. 138 gives an example of an Abelian ring that is not McCoy on a side.

I found the second one using DaRT, but there was no conclusive example for the first one, so I thank you for asking this question.

• Before asking the question, I also searched on DaRT, but didn't find Commented 2 days ago
• After not finding anything on DaRT, my google search for "mccoy ring that is not abelian" had your question as the first hit, and the paper i cite above as the second. Commented 2 days ago
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