Malaysian Journal of Mathematical Sciences, September 2021, Vol. 15, No. 3
On LA-Semimodule Over LA-Semiring
Andari, A. and Rouf, A.
Corresponding Email: ari_mat@ub.ac.id
Received date: 30 October 2019
Accepted date: 30 August 2020
Abstract:
In this paper, we develop an LA-module over LA-ring to a new concept namely LA-semimodule over LA-semiring. Let $S$ be a non-empty set with two binary operations "$+$" and "$\ast$". Set $S$ is called a left almost semiring (LA-semiring) if $(S,+)$ is an LA-semigroup, $(S,\ast)$ is an LA-semigroup and satisfying left and right distributive law of "$\ast$" over "$+$" hold. Let $(S,+,\ast)$ is an LA-semiring with left additive identity equal to $0_S$ and left multiplicative identity equal to 1, non-empty set $M$ is called an LA-semimodule over $S$ if 1) $(M,+)$ is an LA-semigroup with left identity, 2) the map $S \times M \to M,$ $(s,m)\mapsto sm$ where $s \in S$ and $m \in M$ satisfies i) $s(m+n) = sm+sn$, ii) $(r+s)m = rm + sm$, iii) $r(sm)=s(rm)$, iv) $1 \ast m = m$, for all $r,s \in R$, and $m,n \in M$. Then, we investigate the basic properties and the Isomorphims Theorem for LA-semimodule over LA-semiring.
Keywords: LA-semigroup; LA-semiring; LA-semimodule
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