Congruences and External Direct Sum of LA-modules

Fawad Hussain; Muhammad Sajjad Ali Khan; Khaista Rahman  and Madad Khan

doi:10.17485/ijst/2015/v8i28/54260

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Congruences and External Direct Sum of LA-modules

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Indian Journal of Science and Technology

DOI: 10.17485/ijst/2015/v8i28/54260

Year: 2015, Volume: 8, Issue: 28, Pages: 1-7

Original Article

Congruences and External Direct Sum of LA-modules

Fawad Hussain¹, Muhammad Sajjad Ali Khan¹, Khaista Rahman¹ and Madad Khan²

¹ Department of Mathematics, Hazara University, Mansehra, KPK - 21310, Pakistan; [email protected], [email protected], [email protected]
² Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad - 22010, Pakistan; [email protected]

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Abstract In this paper we study a new algebraic structure namely left almost module (LA-module in short).We extend the notion of congruences to LA-modules which is defined in1 for semigroups. We show that every homomorphism defines a congruence relation on LA-modules and prove analogues of isomorphism theorems. We also define external direct sum of LA-modules and show that the internal direct sum of LA-submodules is isomorphic to the external direct sum of those LA-submodules.
Keywords: Congruences, External Direct Sums, Internal Direct Sums, LA-modules, LA-module Homomorphism, LA-submodules