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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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