Semi M-Projective and Semi N-Injective Modules

Abstract

Let $M$ and $N$ be modules over a ring $R$. The purpose of this paper is to study modules $M,N$ for which the bi-module $[M,N]$ is regular or $pi$. It is proved that the bi-module $[M,N]$ is regular if and only if a module $N$ is semi $M-$projective and $\text {Im}(\alpha)\subseteq^{\oplus} N$ for all $\alpha\in [M,N]$, if and only if a module $M$ is semi$N-$injective and $\text {Ker}(\alpha)\subseteq^{\oplus} N$ for all $\alpha\in [M,N]$. Also, it is proved that the bi-module$[M,N]$ is $pi$ if and only if a module $N$ is direct $M-$projective and for any $\alpha\in [M,N]$ there exists $\beta\in [N,M]$ such that $\text {Im}(\alpha\beta)\subseteq^{\oplus} N$, if and only if a module $M$ is direct $N-$injective and for any $\alpha\in [M,N]$ there exists $\beta\in [N,M]$ such that $\text {Ker}(\beta\alpha)\subseteq^{\oplus} M$. The relationship between the Jacobson radical and the (co)singular ideal of $[M,N]$ is described.