The McCoy condition on skew polynomial rings. (English) Zbl 1187.16027

Başer, Muhittin; Kwak, Tai Keun; Lee, Yang

Commun. Algebra 37, No. 11, 4026-4037 (2009).

Let \(R\) be a ring with 1, \(\sigma\) an endomorphism of \(R\), \(R[x]\) the polynomial ring, and \(R[x;\sigma]\) the skew polynomial ring. Then \(R\) is called a \(\sigma\)-skew McCoy ring if for any nonzero \(p(x)=\sum_{i=0}^ma_ix^i\), \(q(x)=\sum_{j=0}^na_jx^j\in R[x;\sigma]\), \(p(x)q(x)=0\) implies \(p(x)c=0\) for some nonzero \(c\in R\). When \(\sigma\) is the identity endomorphism, a \(\sigma\)-skew McCoy ring is a right McCoy ring.
The authors show some properties of a \(\sigma\)-skew McCoy ring. Theorem 1. (1) If \(R[x;\sigma]\) is right McCoy, then \(R\) is \(\sigma\)-skew McCoy, and (2) if \(R\) is \(\sigma\)-skew Armendariz, then \(R\) is \(\sigma\)-skew McCoy, where a ring is called \(\sigma\)-skew Armendariz if for \(p(x)=\sum_{i=0}^ma_ix^i\), \(q(x)=\sum_{j=0}^na_jx^j\in R[x;\sigma]\), \(p(x)q(x)=0\) implies \(a_i\sigma^i(b_j)=0\) for all \(i,j\).
A ring is called reversible if \(ab=0\) implies \(ba=0\) for \(a,b\in R\), and right (left) \(\sigma\)-reversible if \(ab=0\) implies \(b\sigma(a)=0\) (resp., \(\sigma(b)a=0\)) for \(a,b\in R\).
Theorem 2. Let \(\sigma\) be a monomorphism of a reversible ring \(R\) such that \(R\) is right \(\sigma\)-reversible. Then (1) \(R\) is \(\sigma\)-skew McCoy, and (2) for any nonzero \(p(x),q(x)\in R[x;\sigma]\) with \(p(x)q(x)=0\), there exists \(0\neq c\in R\) such that \(cq(x)=0\).
Moreover, it is shown that a \(\sigma\)-skew McCoy ring \(R\) is characterized in terms of some special kinds of upper triangular matrix rings over \(R\) and the polynomial ring \(R[x]\) with the endomorphism \(\overline\sigma\) extended from \(\sigma\), respectively.

Reviewer: George Szeto (Peoria)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16U80	Generalizations of commutativity (associative rings and algebras)
16W20	Automorphisms and endomorphisms

Keywords:

skew McCoy rings; skew polynomial rings; reversible rings; skew Armendariz rings

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