On injectivity and $P$-injectivity
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Abstract
The following results are extended fromP-injectiverings to AP-injective rings: (1)R is left self-injective regular ifand only if R is a right (resp. left)AP-injective ring such that forevery nitely generated leftR-module M , R(M=Z (M )) is projec-tive, whereZ(M ) is the left singular submodule ofRM ; (2) ifR isa left nonsingular leftAP-injective ring such that every maximalleft ideal ofR is either injective or a two-sided ideal ofR, thenR iseither left self-injective regular or strongly regular. In addition, weanswer a question of Roger Yue Chi Ming [13] in the positive. LetRbe a ring whose every simple singular leftR-module isY J-injective.IfR is a rightMI -ring whose every essential right ideal is an essen-tial left ideal, thenR is a left and right self-injective regular, leftand right V-ring of bounded index.
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