International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume V, Issue X, October 2020 | ISSN 2454–6186

 Generalization of Pure-Supplemented Modules

R.S. Wadbude
Mahatma Fule Arts, Commerce and Sitaramji Chaudhari Science Mahavidyalaya, Warud.
SGB Amravati University Amravati [M.S.]

Abstract: Let R be a ring and M be an R-module. We generalized the concepts pure-lifting and pure-supplemented module and introduce weak distribution with fully invariant. We prove every pure g-lifting is pure g-supplemented module. Let M be a weak distribution pure g-supplemented module, then M/A is pure g-supplemented module for every submodule A of M. Let M = M1M2 be a weakly distributive R-module. Then each Mi, i{1, 2}is closed weak g-supplemented if and only if M is closed weak g-supplemented.

Key Words: g-small, g-supplemented, pure-lifting, pure-supplemented, pure g-supplemented, closed weak g-supplemented, Distributive, weak Distributive modules.

I. INTRODUCTION

Throughout this paper R is an associative ring with unity and all modules are unitary R-modules. [12] Sahira M. Yasen and W. Khalid Hasan introduce the concepts pure- module and pure-supplemented module with some conditions. Let M be an R-module, a sub module L of module M is denoted by L ≤ M. submodule L of M is called essential (large) in M, abbreviated K ≤e M, if for every submodule N of M, L ∩ N implies N = 0. A submodule N of a module M is called small in M, denoted by N≪M, if for every sub module L of M, the equality N + L = M implies L = M. [2] A submodule K of m is called generalized small (g-small) submodule of M denoted by N ≪gM, if for every essential submodules T of M with the property M = K + T implies that T = M. Supplemented modules and two other generalizations amply supplemented and weakly supplemented modules were studied by Helmut Zoschinger and he posed their whole structure over discrete valuation rings.

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