Prime Quasi-Ideals in Ternary Seminear Rings

Abstract

Objectives/Background: The ternary seminear ring is the generalization of seminear ring and it need not be a ternary semiring. Characterization of quotient ternary seminear rings and some structures of ternary seminear ring have been analysed and also studied ideals in ternary seminear rings. Further quasi ideals in ternary seminear rings defined and discussed about their properties. Methods: Properties of seminear ring and ternary semiring have been employed to carry out this research work to obtain all the characterizations of ternary seminear rings corresponding to that ternary semiring. Findings: We call an algebraic structure (T;+; :) is a ternary seminear ring if (T,+) is a Semigroup, T is a ternary semigroup under ternary multiplication and xy(z+u) = xyx+xyu for all x;y; z;u 2 T. T is said to have an absorbing zero if there exists an element 0 2 T such that x+0 = 0+x = x for all x 2 T and xy0 = x0y = 0xy = 0 for all x;y 2 T. Throughout this paper T will always stand for ternary seminear ring with an absorbing zero. In this ternary structure we try to study prime quasi ideals concept and obtain their properties. Novelty: In this study, we define the notion of Prime quasi ideals in ternary seminear rings. We also find some of their interesting results. AMS Subject Classification code: 16Y30,16Y99,17A40 Keywords: Ternary seminear ring; Idempotent ternary seminear ring; Ideals in ternary seminear ring; Quasi Ideals in ternary seminear ring; Prime Ideals in ternary seminear rings.