Rings with restricted minimum condition

Abstract

Definition. A ring R is said to satisfy the restricted minimum condition (or to be a RM ring, for short), if for each ideal A $ (0) in R, the ring R/A is right artinian. In this paper we consider a RM ring, and are furthermore interested in the case where R itself is not right artinian. The concept of a commutative RM ring was introduced by I. S. Cohen [1], who also proved the following results for a commutative ring R: (a) R is RM iff R is noetherian and every proper prime ideal is maximal. (b) R is RM but not artinian iff R is a noetherian integral domain not a field, in which every proper prime ideal is maximal.