A study on dual square free modules

Let $M$ be an $H$-supplemented coatomic module with FIEP. Then we prove that $M$ is dual square free if and only if every maximal submodule of $M$ is fully invariant. Let $M=\bigoplus_{i\in I} M_i$ be a direct sum, such that $M$ is coatomic. Then we prove that $M$ is dual square free if and only if each $M_i$ is dual square free for all $i\in I$ and, $M_i$ and $\bigoplus_{j\neq i}M_j$ are dual orthogonal. Finally we study the endomorphism rings of dual square free modules. Let $M$ be a quasi-projective module. If $\operatorname{End}_R(M)$ is right dual square free, then $M$ is dual square free. In addition, if $M$ is finitely generated, then $\operatorname{End}_R(M)$ is right dual square free whenever $M$ is dual square free. We give several examples illustrating our hypotheses.