Dual Pure Rickart Modules and Their Generalization
Abstract: 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.
Keywords: dual Pure Rickart modules, dual Pure -Rickart modules, regular modules, Pure submodules, relatively dual Rickart modules
How to Cite?: Ghaleb Ahmed, "Dual Pure Rickart Modules and Their Generalization", Volume 6 Issue 2, February 2017, International Journal of Science and Research (IJSR), Pages: 882-886, https://www.ijsr.net/getabstract.php?paperid=ART2017663, DOI: https://dx.doi.org/10.21275/ART2017663