In this paper, we introduce and study a torsion-theoretic generalization of the weakly-flat module using the concept of \(\tau \)-closed submodule for an idempotent radical \(\tau \). We introduce a new class of modules called weakly-\(\tau \)-flat. We study these modules, generalizing several results both on extending modules and flat modules. We study generalizations and characterizations of IF ring, (finitely) \(\varSigma \)-CS ring, and C-ring by weakly-\(\tau \)-flat modules.

中文翻译：

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弱-$$ \ tau $$τ-平面模块

在本文中，我们针对幂等根\（\ tau \）使用\（\ tau \）-封闭子模块的概念，介绍和研究了弱平坦模块的扭转理论推广。我们引入了一类新的模块，称为弱- \（\ tau \）- flat。我们研究了这些模块，概括了扩展模块和平面模块的一些结果。我们研究了IF环，（有限地）\（\ varSigma \）- CS环和C-环的一般化和特征，它们是通过- \（\ tau \）- flat模块来实现的。