Some Results on delta-Semiperfect Rings and delta-Supplemented Modules
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In [9], the author extends the definition of lifting and supplemented modules to δ-lifting and δ-supplemented by replacing "small submodule" with "δ-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of δ-lifting and δ-supplemented modules. Especially, we show that any finite direct sum of delta-hollow modules is delta-supplemented. On the other hand, the notion of amply δ-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply delta-supplemented and satisfies Descending Chain Condition (DCC) on δ-supplemented modules and on δ-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a δ-semiperfect ring which satisfies DCC on delta-small right ideals of R.
Abstract
In [9], the author extends the definition of lifting and supplemented modules to δ-lifting and δ-supplemented by replacing "small submodule" with "δ-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of δ-lifting and δ-supplemented modules. Especially, we show that any finite direct sum of delta-hollow modules is delta-supplemented. On the other hand, the notion of amply δ-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply delta-supplemented and satisfies Descending Chain Condition (DCC) on δ-supplemented modules and on δ-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a δ-semiperfect ring which satisfies DCC on delta-small right ideals of R.
참고문헌 (13)
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