Difference Ordered G􀀀 Semirings

Objectives: In this study, we generalize some of the difference ordered and weak uniquely difference ordered semirings results. Methods: To establish the results in semirings, we use conditions like commutativity, simple, multiplicative cancellative, additively idempotent on difference-ordered, and weak uniquely difference-ordered semirings. Findings: First, we give some examples of difference ordered semiring, and weak difference ordered semirings. Then generalize some of the results of semirings to semirings and discuss some of the properties of additive idempotent semifield. Novelty: We find that if is a non-zeroic difference ordered semiring then is a strong ideal of Let be a positive difference ordered Gel’fand semiring then every maximal non-unit of is prime. Further, we find that if is a simple difference ordered additively idempotent semiring and which is not a unit then is prime if and only if there exists a character : satisfying Moreover, if and are distinct prime elements of Then and are also distinct. Finally, we consider some properties of additive idempotent semifield and then introduce the concept of weak uniquely difference-ordered semirings. AMS Mathematics subject classification (2020): 16Y60. Keywords: Γ-semiring, Difference ordered Γ-semiring, Additively idempotent Γ- semiring, Strong identity, Weak uniquely difference-ordered Γ-semirings