Power Series Rings Satisfying a Zero Divisor Property

Abstract

In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B 1,…,B m , such that B 1… B m = 0 and B i = A i [[x]] for some minimal prime ideal A i of R for all i, where A 1,…,A m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.