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Semi-commutativity and the McCoy condition. (English) Zbl 1110.16036
A ring \(R\) is called ‘right McCoy’ if it satisfies: whenever \(f(x)g(x)=0\) in \(R[X]\) for non-zero polynomials \(f(x)\) and \(g(x)\), then there is a non-zero \(r\in R\) with \(f(x)r=0\). \(R\) is called a ‘McCoy ring’ if it is both left and right McCoy. It is known that any commutative ring and any reduced ring is a McCoy ring. A natural question is whether there is a class of McCoy rings which includes both the commutative rings and the reduced rings. The author shows that the class of reversible rings fulfils this requirement (\(R\) is ‘reversible’ if \(ab=0\) implies \(ba=0\) for \(a,b \in R\)). This confirms the earlier unpublished result to this effect of T. Y. Lam, A. Leroy and J. Matczuk.
The well-known open question on whether the more general semi-commutative rings (\(ab=0\) implies \(aRb=0\) with \(a,b\in R\)) are McCoy rings, which was elsewhere claimed to be true, is also settled here in the negative: an example of a semi-commutative ring is given which is not McCoy. It is shown that the semi-commutative rings satisfy a weaker form of the McCoy condition.

16U80 Generalizations of commutativity (associative rings and algebras)
16S36 Ordinary and skew polynomial rings and semigroup rings
Full Text: DOI
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