Prime Bi-Ternary �?Ž�?�??-Ideals in Ternary �?Ž�?�??-Semirings

This paper is divided into four sections, in section 1, introduction of the paper. In section 2, we provided some preliminaries which are useful for further development of the paper. In section 3, we were introduced bi-ternary Γ-ideals, strongly bi-ternary Γ-ideals and semiprime bi-ternary Γ- ideals in ternary Γ-semirings. It is proved that (1) Every strongly prime bi-ternary Γ-ideal of a ternary Γ-semiring T is a prime bi-ternary Γ-ideal of T. (2) Every prime bi-ternary Γ-ideal of a ternary Γ-semiring T is a semiprime bi-ternary Γ-ideal of T. In section 4, the terms irreducible and strongly irreducible bi-ternary Γ-ideals in ternary Γ-semiring. It is proved that if B be a biternary Γ-ideal of a ternary Γ-semiring T and a ∈ T such that a∉ B. Then there exists an irreducible bi-ternary Γ-ideal I of T such that B ⊆ I and a ∉ I.