On rings with restricted minimum condition (Q1110632)

An associative ring R with unit element is called right RA if every cyclic right R-module is artinian, and it is called right CPA if every cyclic right R-module is a direct sum of a projective module and an artinian module. Thus, a right CPA ring satisfies the restricted minimum condition on the right, that is, R/A is artinian for every essential right ideal A. The authors prove that a right CPA ring is the module theoretical direct sum of an artinian ideal and finitely many uniform right ideals. They give a description of the structure of these rings for the case when their right socle is zero. While it remains open whether a right CPA ring R is always noetherian (this is not even known for the more special case of a non-artinian right RA ring), it is shown that right hereditary right CPA rings are right noetherian. Furthermore, a two-sided noetherian hereditary ring is left and right CPA.