Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right $H$-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

中文翻译：

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在简单直接模块上

最近，在一系列论文中，已经研究了直接注入和直接投影模块的“简单”版本。这些模块分别被称为“简单直接射影”和“简单直接射影”。在本文中，我们在整数环和半局域环上给出了上述模块的完整表征。环是半局域的，当且仅当每个具有零 Jacobson 根的右模都是简单直接射影的。简单直接射右模是简单直接射影的环被充分表征。这些正是左完美右 $H$ 戒指。其简单直接射影右模是简单直接射影的环是右最大环。对于可交换的 Noetherian 环，我们证明 simple-direct-projective 模块是 simple-direct-injective 当且仅当 simple-direct-injective 模块是 simple-direct-projective 当且仅当环是 Artinian 时。给出了各种闭包属性和一些简单直接注入（分别是投影）的模块类。