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Extensions of zip rings. (English) Zbl 1221.16019
Following C. Faith [Publ. Mat., Barc. 33, No. 2, 329-338 (1989; Zbl 0702.16015)], an associative ring $$R$$ with unit element is called a right zip ring if, given a subset $$\emptyset\neq X\subseteq R$$ with right annihilator $$r(X)=0$$, there exists a finite subset $$Y\subseteq X$$ with $$r(Y)=0$$.
In the article under review, the author investigates under what circumstances this zero intersection property is inherited by certain extension rings. The main results of the paper are as follows. (1) A right McCoy ring $$R$$ is right zip if and only if the polynomial ring $$R[x]$$ is right zip. (2) If $$R$$ is reversible (i.e., $$ab=0$$ implies $$ba=0$$ for all $$a,b\in R$$) and $$M$$ is a unique product monoid (i.e., given any two finite subsets $$\emptyset\neq A,B\subseteq M$$, there exists an element of $$AB$$ having a unique representation $$ab$$, where $$a\in A$$, $$b\in B$$), then $$R[M]$$ is right zip if and only if $$R$$ is right zip.

##### MSC:
 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16S36 Ordinary and skew polynomial rings and semigroup rings 16S50 Endomorphism rings; matrix rings 16U80 Generalizations of commutativity (associative rings and algebras) 16D25 Ideals in associative algebras 20M25 Semigroup rings, multiplicative semigroups of rings
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