PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 33(47), pp. 239245 (1983) 

ON QUASIFROBENIUSEAN AND ARTINIAN RINGSRoger Yue Chi MingUniversité Paris VII, U.E.R. de Mathématique et Informatique, 75251 Paris Cedex os, FranceAbstract: Left $p$injective rings, which extend left self injective rings, have been considered in several papers (cf.\ for example, [10]  [14]). The following generalizations of left $p$injective rings are here introduced: (1) $A$ is called a left mininjective ring if, for any minimal left ideal $U$ of $A$ (if it exists), any left $A$homomorphism $g: U\to A$, there exists $y\in A$ such that $g(b)= by$ for all $b\in U$; (2) $A$ is left $np$injective if, for any nonnilpotent element $c$ of $A$, any left Ahomomorphism $g: Ac\to A$, there exists $y\in A$ such that $g(ac)= acy$ for all $a\in A$. New characteristic properties of quasiFrobeniusean rings are given. It is proved that A is quasiFrobeniusean iff $A$ is a left Artinian, left and right mininjective ring. If $A$ is left $np$injective, then (a) every left or right $A$module is divisible and (b) any reduced principal left ideal of $A$ is generated by an idempotent. Further properties of left $CM$rings (introduced in [14]) are developed. The following nice result is established : If $U$ is a minimal left ideal of a left $CM$ring $A$, the following are then equivalent: (a) $_AU$ is injective; (b) $_AU$ is projective; (c) $_AU$ is $p$injective. Consequently, $A$ is semisimple Artinian iff $A$ is a left $CM$ring with finitely generated projective essential left socle. Divison rings are also characterised. Known results are improved. Keywords: quasiFrobeniusean, Artinian, von Neumann regular, mininjective, npinjective, CMring Classification (MSC2000): 16A30, 16A35, 16A36, 16A40, 16A52 Full text of the article:
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