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A note on zip rings. (English) Zbl 1295.16016
A ring \(R\) is called right zip if the right annihilator \(r_R(S)=0\) of a subset \(S\subseteq R\) then \(r_R(X)=0\) for any finite subset \(X\subseteq S\). The aim of the paper is to show that to the list of rings \(R\) which being right zip imply that monoidal rings \(R[M]\) are zip for some classes of monoids \(M\), the class of uniform rings can be added. The author, motivated by E. Hashemi [Stud. Sci. Math. Hung. 47, No. 4, 522-528 (2010; Zbl 1221.16019)], shows that a uniform ring \(R\) is right zip if and only if \(R[M]\) is right zip provided \(M\) is a u.p.-monoid (unique product monoid). Then, some conclusions are derived.

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