Abstract In this paper, we introduce the concept of δss-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δss-supplemented as a left module if and only if RSoc(RR) {R \over {Soc\left( {_RR} \right)}} is semisimple and idempotents lift to Soc(RR) if and only if every left R-module is δss-supplemented. We define projective δss-covers and prove the rings with the property that every (simple) module has a projective δss-cover are δss-supplemented. We also study on δss-supplement submodules.

中文翻译：

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δss 补充模块和环

摘要 在本文中，我们介绍了 δss 补充模块的概念，并提供了这些模块的各种特性。特别地，我们证明了一个环 R 是 δss 补充为左模块当且仅当 RSoc(RR) {R \over {Soc\left( {_RR} \right)}} 是半简单的并且幂等项提升到 Soc( RR) 当且仅当每个左 R 模都是 δss 补充的。我们定义了射影 δss-covers 并证明了具有每个（简单）模块具有射影 δss-cover 的性质的环是 δss 补充的。我们还研究了 δss-supplement 子模块。