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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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