论文标题
在简单的直接模块上
On Simple-Direct Modules
论文作者
论文摘要
最近，已经研究了一系列“简单”版本的直接注射和直接标记模块的论文。这些模块分别称为“简单直接注射”和“简单的直接标记”。在本文中，我们对整数环和半局部环上的上述模块进行了完整的表征。当且仅当每个右边的雅各布森自由基为零的模块都是简单的独立训练时，该环是半遗传的。简单直接注射右模块的环是简单的直接指控性的。这些正是左右$ H $ rings。简单直接指定右右模块的环是简单的注射式注射词，是正确的最大环。对于换向的Noetherian环，我们证明，当且仅当戒指是Artinian时，仅当简单的独立注射模块才是简单的独立注射模块时，简单的直接指定模块才是简单的直接注射。给出了各种闭合属性和一些简单直接注射的模块（分别投影）。
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.
