Extensions of zip rings

Abstract

A ring R is called right zip provided that if the right annihilator rR(X) of a subset X of R is zero, then there exists a finite subset Y of X, such that rR(Y) = 0. Faith [6] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; When does R being a right zip imply R[G] being right zip when G is a finite group?; Characterize a ring R such that Matn(R) is right zip. In this note we continue the study of the extensions of non-commutative zip rings based on Faith’s questions. It is shown that if R is a right McCoy ring, then R is right zip if and only if R[x] is a right zip ring. Also, if M is a strictly totally ordered monoid and R a right duo ring or a reversible ring, then R is right zip if and only if R[M] is right zip. As a consequence we obtain a generalization of [7].