summary:Let $S$ and $R$ be two associative rings, let $ _{S}C_{R}$ be a semidualizing $(S,R)$-bimodule. We introduce and investigate properties of the totally reflexive module with respect to $_{S}C_{R}$ and we give a characterization of the class of the totally $C_{R}$-reflexive modules over any ring $R$. Moreover, we show that the totally $C_{R}$-reflexive module with finite projective dimension is exactly the finitely generated projective right $R$-module. We then study the relations between the class of totally reflexive modules and the Bass class with respect to a semidualizing bimodule. The paper contains several results which are new in the commutative Noetherian setting.
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机译：摘要：让$ S $和$ R $为两个关联环，让$ _ {S} C_ {R} $为半对偶$（S，R）$-bimodule。我们介绍和研究关于$ _ {S} C_ {R} $的完全自反模块的属性，并给出了在任何环$ R $上的完全$ C_ {R} $自反模块的类的特征。此外，我们证明了具有有限射影维度的完全$ C_ {R} $-自反模块恰好是有限生成的射影右$ R $-模块。然后，我们针对半对偶双模块研究了完全自反模块的类与Bass类之间的关系。本文包含了一些交换Noetherian设置中的新结果。
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