Lifting modules with respect to images of a fully invariant submodule

Tayyebeh Amouzegar, Ali Reza Moniri Hamzekolaee

Lifting modules as a main concept in module theory have been studied and investigated extensively in recent decades. The first author in \cite{amoozegar} tried to consider and investigate this concept with a homological approach. Let $R$ be a ring and $M$ be a right $R$-module. Then $M$ is called $\mathcal{I}$-lifting if image of every endomorphism of $M$ lies above a direct summand of $M$. In this paper, we are interested to study module $M$ with this property that $\varphi(F)/D\ll M/D$ for every endomorphism $\varphi$ of $M$ and for some direct summands $D$ of $M$, where $F$ is a fixed fully invariant submodule of $M$. We call such modules $\mathcal{I}_F$-lifting. We provide some examples of $\mathcal{I}_F$-lifting modules as a proper generalization of lifting modules. Some characterizations of $\mathcal{I}_F$-lifting modules are given. We also define relative $\mathcal{I}_F$-lifting modules to study direct summands and finite direct sums of $\mathcal{I}_F$-lifting modules.