Solution of an open problem on cofaithful one-sided ideals

Abstract

We solve the following open problem in the negative: does the power series ring R[[x]] of a ring R inherit from R the property of every faithful right ideal being cofaithful? In other words, we construct a ring R such that every faithful right ideal of R is cofaithful, i.e., whenever the right annihilator rannR(K) of a right ideal K in R is {0}, then rannR(Y)={0} for some non-empty finite subset Y of K, but R[[x]] does not have the mentioned property.