In this paper we introduce dual $F$-Baer modules and give a characterization of them. Let $M$ be a module and $F$ a fully invariant submodule of $M$. $M$ is called dual $F$-Baer if for every family of endomorphisms ${\{g_{\alpha }\}}_{\alpha \in I}$ of $M$, ${\sum }_{\alpha \in I}{g}_{\alpha }(F)$ is a direct summand of $M$. We prove that $M$ is dual $F$-Baer if and only if $M=F\oplus N$ for some submodule $N$ of $M$ with $F$ dual Baer. We obtain a positive solution for the Schröder–Bernstein problem for certain dual $F$-Baer modules.

中文翻译：

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双 F-Baer 模块的 Schröder-Bernstein 问题

在本文中，我们介绍了对偶$F$-Baer 模块并给出它们的特征。让$米$成为一个模块并且$F$的一个完全不变的子模块$米$.$米$被称为双$F$-Baer if 对于每一个自同态家族${\{G_{\alpha }\}}_{\alpha \in 我}$的$米$,${\sum }_{\alpha \in 我}{G}_{\alpha }(F)$是直接总和$米$. 我们证明$米$是双重的$F$-Baer 当且仅当$米=F\oplus ñ$对于一些子模块$ñ$的$米$和$F$双贝尔。对于某些对偶，我们获得了 Schröder-Bernstein 问题的正解$F$-贝尔模块。