Some results on quasi-t-dual Baer modules

Let $R$ be a ring and let $M$ be an $R$-module with $S=\rm{End}_R(M)$. Consider the preradical ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}$ for the category of right $R$-modules Mod-$R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}(M) = \bigcap \{U\leq M\colon M/U$ is small in its injective hull$\}$. The module $M$ is called quasi-t-dual Baer if $\sum_{\varphi \in \mathfrak{I}} \varphi({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M))$ is a direct summand of $M$ for every two-sided ideal $\mathfrak{I}$ of $S$, where ${{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) = {{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}} ({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}(M))$. In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) $ is a direct summand of $M$ and ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}^2(M)$ is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.