The McCoy condition on noncommutative rings. (English) Zbl 1231.16032

Hong, Chan Yong; Jeon, Young Cheol; Kim, Nam Kyun; Lee, Yang

Commun. Algebra 39, No. 5, 1809-1825 (2011).

Let \(R\) be a ring with 1 and \(R[x]\) a polynomial ring with indeterminate \(x\). Then \(R\) is called right McCoy if for nonzero \(f(x)\), \(g(x)\) in \(R[x]\), \(f(x)g(x)=0\) implies \(f(x)r=0\) for some nonzero \(r\in R\). Left McCoy rings and McCoy rings are similarly defined. A ring is called strongly right McCoy if \(f(x)g(x)=0\) implies \(f(x)r=0\) for some nonzero \(r\) in the right ideal of \(R\) generated by the coefficients of \(g(x)\), where \(f(x)\neq 0\), \(g(x)\neq 0\) in \(R[x]\). A left strongly McCoy ring and a strongly McCoy ring are similarly defined. Strongly right McCoy rings are right McCoy, but the converse is not true. Relationships are given between strongly McCoy rings and other classes of rings such as Abelian rings (i.e., every idempotent is central) and duo rings (i.e., every one sided ideal is two sided). Moreover, relationships are given between strongly right McCoy rings and some extension rings.
Theorem 1. Let \(R\) be a right Ore ring with its classical right quotient ring \(Q_r(R)\). Then \(R\) is strongly right McCoy if and only if so is \(Q_r(R)\). Theorem 2. Let \(D_n(R)\) be the ring of all \(n\times n\) upper triangular matrices with a constant main diagonal, and \(V_n(R)\subset D_n(R)\) with all constant diagonals parallel to the main diagonal. Then the following conditions are equivalent: (1) \(R\) is strongly right (resp. left) McCoy; (2) \(D_2(R)\) is strongly right (resp. left) McCoy; (3) \(V_n(R)\) is strongly right (resp. left) McCoy for any \(n\geq 2\). Theorem 3. If \(R\) is right duo or reversible (i.e., \(ab=0\) implies \(ba=0\) for \(a,b\in R\)), then \(R[x]\) is strongly right McCoy.

Reviewer: George Szeto (Peoria)

MSC:

16U80	Generalizations of commutativity (associative rings and algebras)
16S36	Ordinary and skew polynomial rings and semigroup rings
16S50	Endomorphism rings; matrix rings

Keywords:

classical quotient rings; McCoy theorem; polynomial rings; reversible rings; right duo rings; strongly right McCoy rings

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