FULLY INVARIANT τM-LIFTING MODULES

Abstract

Let ${\tau }_M$ be any preradical for $\sigma \left[M\right]$ and $N$ any module in $\sigma \left[M\right]$. A module $N$ is called ${\tau }_M$-lifting if for every submodule $K$ of $N$, there is a decomposition $K=A\oplus B$, such that $A$ is a direct summand of $N$ and $B\subseteq {\tau }_M\left(N\right)$. We call $N$ is (strongly) $FI‐{\tau }_M$-lifting if for every fully invariant submodule $K$ of $N$, there is a decomposition $K=A\oplus B$, such that $A$ is a (fully invariant) direct summand of $N$ and $B\subseteq {\tau }_M\left(N\right)$. The class of $\mathrm{FI}‐{\tau }_M$-lifting modules properly contains the class of ${\tau }_M$-lifting modules and the class of strongly $\mathrm{FI}‐{\tau }_M$-lifting modules. In this paper we investigate whether the class of (strongly) $\mathrm{FI}‐{\tau }_M$-lifting modules are closed under particular class of submodules, direct summands and direct sums.