A note on semi-prime rings
The following generalization of semi-prime rings is introduced: A is called an SRN ring if the sum of all reduced ideals of A contains no non- zero nilpotent ideal of A. (An ideal is reduced iff it contains no non- zero nilpotent element.) It is proved that A is SRN iff the sum of all reduced ideals of A is a reduced ideal. The unique maximal strongly regular ideal of some SRN rings is then considered. If A is semi-prime, then (a) any complement left ideal which is an ideal is a left annihilator; (b) if A is also left CM [see the author, ibid. 91, 153-166 (1981; Zbl 0452.16007)], then any complement left ideal is a left annihilator. Certain properties of semi-prime right self-injective rings are given. If A is right self-injective (resp. continuous) regular such that for any proper complement left ideal C, either C is an ideal of A or r(C)\(\neq 0\), then A is left self-injective (resp. continuous). Simple Artinian and semi-simple Artinian rings are characterized.
scientific article; zbMATH DE number 3941724