Indian Journal of Science and Technology

Abstract

Objectives: The main objective of this paper is to derive some of the results of k􀀀irreducible ideals, common right divisors and Euclidean G 􀀀 semiring: Methods: To establish the main results in G 􀀀 semirings, we use some conditions like commutativity, simple, semi subtractive, centreless, multiplicative G􀀀idempotent, strong multiplicative G􀀀idempotent, additively cancellation and the concept of common right divisor and Euclidean norm. Findings: First we study some results regarding k􀀀irreducible ideals and define a a􀀀 generated ideal by any element of R. In connection with different conditions, we characterize some results of irreducibility in ideals, primary ideals, common right divisors and Euclidean norms. Novelty: Ana􀀀 generated ideal by any element of R, say a, denoted by <aa > is k􀀀ideal if R is simple, semi subtractive and additive cancellative. Again, the conditions mentioned above are used to prove that every k􀀀irreducible ideal of a G􀀀semiring R is primary ideal of R. Furthermore, the concept of Euclidean G􀀀semiring developed to establish various results in the theory of G􀀀semirings.

Keywords: k-Irreducible Ideal; Common Right Divisor; Euclidean Norm; Primary Ideals; Noetherian G-Semirings