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

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.

### MSC:

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

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