A NOTE ON YJ-INJECTIVITYIn [ll],p-injective modules are introduced to study von Neumann regular rings and associated rings. In [13], this concept is weakened to Y J-injectivity. Recall that (1) A left A-module M is p-injective if, for any principal left ideal P of A, every left A-homomorphism of P into M extends to A;(2) AM is Y J-injective if, for any 0 / a S A, there exist a positive integer n (depending on a) such that a n ^ 0 and every left Ahomomorphism of We write «A is MELT» if any maximal essential left ideal of A (if it exists) is an ideal of A. MELT rings generalize effectively left quasi-duo rings and semi-simple Artinian rings.In this note, we consider various rings whose simple one-sided modules are Y J-injective. This is motivated by a well-known theorem of I. Kaplansky: A commutative ring A is von Neumann regular iff every simple A module is injective.
Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.
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