Module decompositions via Rickart modules

This work is devoted to the investigation of module decompositions which arise from Rickart modules, socle and radical of modules. In this regard, the structure and several illustrative examples of inverse split modules relative to the socle and radical are given. It is shown that a module $M$ has decompositions $M=\operatorname{Soc}(M) \oplus N$ and $M=\operatorname{Rad}(M) \oplus K$ where $N$ and $K$ are Rickart if and only if $M$ is $\operatorname{Soc}(M)$-inverse split and $\operatorname{Rad}(M)$-inverse split, respectively. Right $\operatorname{Soc}(\,\cdot\,)$-inverse split left perfect rings and semiprimitive right hereditary rings are determined exactly. Also, some characterizations for a ring $R$ which has a decomposition $R=\operatorname{Soc}(R_R)\oplus I$ with $I$ a hereditary Rickart module are obtained.