ON QUASI-FROBENIUSEAN AND ARTINIAN RINGS

Roger Yue Chi Ming

Abstract: 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 min-injective 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 non-nilpotent element $c$ of $A$, any left A-homomorphism $g: Ac\to A$, there exists $y\in A$ such that $g(ac)= acy$ for all $a\in A$. New characteristic properties of quasi-Frobeniusean rings are given. It is proved that A is quasi-Frobeniusean iff $A$ is a left Artinian, left and right min-injective 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 semi-simple 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: quasi-Frobeniusean, Artinian, von Neumann regular, min-injective, np-injective, CM-ring

Classification (MSC2000): 16A30, 16A35, 16A36, 16A40, 16A52

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