PS-lifting modules

Abstract

UDC 512.55

Let $R$ be a ring and let  $M$ be a left $R$-module. We say that $M$ is {\it ps-lifting} if every submodule $N$ of $M$  contains a direct summand $X$ of $M$ such that $\dfrac{N}{X}$ is projective semisimple. We present some properties of these modules. It is shown that: (1) if a projective module is ps-lifting, then it is hereditary; (2) for a ring $R,$ every left $R$-module is ps-lifting if and only if every $R$-module is a direct sum of an injective module and a projective semisimple module; (3) $_{R}R$ is ps-lifting if and only if $\dfrac{R}{\rm Soc(R)}$ is semisimple and $R$ is hereditary.