Dual Pure Rickart Modules and Their GeneralizationJournal: International Journal of Science and Research (IJSR) (Vol.6, No. 2)
Publication Date: 2017-02-05
Authors : Ghaleb Ahmed;
Page : 882-886
Keywords : dual Pure Rickart modules; dual Pure -Rickart modules; regular modules; Pure submodules; relatively dual Rickart modules;
Let R be a commutative ring with identity and M be an R-module. In this paper we introduce the dual concepts of Pure Rickart modules and Pure -Rickart modules as a generalization of dual Rickart modules and dual -Rickart modules respectively. Further, dual Pure Rickart modules and dual Pure -Rickart modules can be considered as a generalization of regular rings and -regular rings respectively. Furthermore, dual Pure -Rickart modules is a generalization of Pure Rickart modules. An R-module M is called dual Pure Rickart if for every f EndR (M), Im f is a pure (in sense of Anderson and Fuller) submodule of M. An R-module M is called dual Pure - Rickart if for every f EndR (M), there exist a positive integer n such that Im f^n is a pure (in sense of Anderson and Fuller) submodule of M. We show that several results of dual Rickart modules and dual -Rickart modules can be extended to dual Pure Rickart modules and dual Pure -Rickart modules for this general settings. Many results about these concepts are given and some relationships between these modules and other related modules are investigated.
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