On the translational invariant intuitionistic fuzzy subset of a Γ-ring

In this paper, we introduce the notion of translational invariant intuitionistic fuzzy subset of a Γ-ring and generalize some notions of a ring to a Γ-ring. Also, we define ideals of a Γ-ring generated by an intuitionistic fuzzy subset with an element of Γ-ring and study their properties. The notion of units, associate, prime element, irreducible element are also generalized with respect to the intuitionistic fuzzy subset of a Γ-ring. Further, we study the properties of homomorphic image and pre-image of translational invariant intuitionistic fuzzy subset under the Γ-ring homomorphism and we prove that every homomorphic image of a prime ideal of a Γ-ring generated by an Aγ-prime element and translational invariant and f-invariant intuitionistic fuzzy subset is also a prime ideal.