On Simple Ternary 𝚪-Semiring

In this paper, the terms, simple ternary Γ-semiring, semi-simple, semisimple ternary Γ-semiring are introduced.It is proved that (1) If T is a left simple ternary Γ-semiringor a lateral simple ternary Γ-semiring or a right simple ternary Γ-semiring then T is a simple ternary Γ-semiring.(2) A ternary Γ-semiring T is simple ternary Γsemiring if and only if TΓTΓaΓTΓT = T for all a T. (3) A ternary Γ-semiring T is regular then every principal ternary Γ-ideal of T is generated by an idempotent.(4) An element a of a ternary Γ-semiring T is said to be semi simple if aLet T be a ternary Γ-semiring and aT  .If a is regular, then a is semisimple.(17) a be an element of a ternary Γ-semiringT and a is left regular or lateral regular or right regular, then a is semisimple.(18) Let a be an element of a ternary semiring T and if a is intra regular then a is semisimple.