Indian Journal of Science and Technology

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Congruences and External Direct Sum of LA-Modules


Affiliations
1 Department of Mathematics, Hazara University, Mansehra, KPK - 21310, Pakistan
2 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad - 22010, Pakistan
 

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-Sub Modules
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  • Congruences and External Direct Sum of LA-Modules

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Authors

Fawad Hussain
Department of Mathematics, Hazara University, Mansehra, KPK - 21310, Pakistan
Muhammad Sajjad Ali Khan
Department of Mathematics, Hazara University, Mansehra, KPK - 21310, Pakistan
Khaista Rahman
Department of Mathematics, Hazara University, Mansehra, KPK - 21310, Pakistan
Madad Khan
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad - 22010, Pakistan

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-Sub Modules



DOI: https://doi.org/10.17485/ijst%2F2015%2Fv8i28%2F121378