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Extensions of zip rings. (English) Zbl 1071.16020
Throughout \(R\) is an associative ring with identity. Denote by \(UTM_n(R)\) the \(n\times n\) upper triangular matrix ring over a ring \(R\) and by \(aUTM_n(R)\) (\(n\geq 2\)) the ring having as elements upper triangular matrices with entries in \(R\) and all the entries on the main diagonal equal. A ring \(R\) is called right zip provided that if the right annihilator \(r_R(X)\) of a subset \(X\) of \(R\) is zero, then \(r_R(Y)=0\) for a finite subset \(Y\subseteq X\).
C. Faith raised [in Commun. Algebra 19, No. 7, 1867-1892 (1991; Zbl 0729.16015)] a number of questions on zip rings, in connection with which the authors establish a series of results. It is proved that a ring \(R\) is a right (left) zip ring if and only if \(aUTM_n(R)\) is a right (left) zip ring. An Armendariz ring \(R\) is a right zip ring if and only if \(R[x]\) is a right zip ring.
Given a monoid \(G\) and a ring \(R\), denote by \(R[G]\) the monoid ring of \(G\) over \(R\). A monoid \(G\) is called a u.p. monoid (unique product monoid) if given any two non-empty finite subsets \(A\) and \(B\) of \(G\) there exists at least one \(c\in G\) that has a unique representation in the form \(c=ab\) with \(a\in A\) and \(b\in B\). The authors show that, for a commutative ring \(R\) and a u.p. monoid \(G\) that contains an infinite cyclic submonoid, \(R\) is a zip ring if and only if \(R[G]\) is a zip ring.

16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S36 Ordinary and skew polynomial rings and semigroup rings
16S50 Endomorphism rings; matrix rings
16D25 Ideals in associative algebras
16S34 Group rings
20M25 Semigroup rings, multiplicative semigroups of rings
Full Text: DOI
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