Bull. Korean Math. Soc. 2006 Vol. 43, No. 2, 299-307 Published online June 1, 2006

Guangshi Xiao and Wenting Tong Nanjing University of Aeronautics and Astronautics, Nanjing University

Abstract : 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 finitely generated left $R$-module $M$, $_{R}(M/Z(M))$ is projective, where $Z(M)$ is the left singular submodule of $_{R}M$; (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 $YJ$-injective. If $R$ is a right $MI$-ring whose every essential right ideal is an essential left ideal, then $R$ is a left and right self-injective regular, left and right $V$-ring of bounded index.