The following results are extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every flnitely generated left R-module M, R(M=Z(M)) is projec- tive, where Z(M) is the left singular submodule of RM; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming (13) in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essen- tial left ideal, then R is a left and right self-injective regular, left and right V -ring of bounded index.