Fawad Hussain ¹, Muhammad Sajjad Ali Khan ¹, Khaista Rahman ¹, Madad Khan ²

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

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

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Fawad Hussain ¹, Muhammad Tahir ², Saleem Abdullah ², Nazia Sadiq ²

Affiliations
1 Department of Mathematics, Abbottabad University of Science and Technology, PK
2 Department of Mathematics, Hazara University Mansehra, PK

Abstract

The study of a seminear-ring was started in 1967. Seminear-ring is the generalization of semiring and nearing. It is known that in a semiring the quotient structure is constructed by congruence relations through c-ideals and homomorphisms. We apply the said congruence relations to a seminear-ring and get quotient structure of a seminear-ring. The aim of this paper is to discuss quotient seminear-rings in two different ways. We study congruence relations, homomorphisms and ideals. We show that each homomorphism and c-ideal define a congruence relation on seminear-rings. At the end, we discuss quotient seminear-rings.

Keywords

C-Ideals, Congruence Relation, Homomorphism, Seminear-Ring, Quotient Seminear-Ring.

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