The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. As an extension of intuitionistic fuzzy normed rings, we define the concept of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals. Some essential operations specially subset, complement, union, intersection and several properties relating to the notion of generalized intuitionistic fuzzy normed rings are identified. Homomorphism and isomorphism of intuitionistic fuzzy normed subrings are characterized. We identify the image and the inverse image of intuitionistic fuzzy normed subrings under ring homomorphism and study their elementary properties. Some properties of intuitionistic fuzzy normed rings and relevant examples are presented.

In this paper, we define direct product of finite intuitionistic fuzzy normed normal subrings and investigate relevant properties. We also provide some essential operations specially subset, complement and intersection related to direct product of finite intuitionistic fuzzy normed normal subrings. Then we generalize the relation between the intuitionistic characteristic function and direct product of finite intuitionistic fuzzy normed normal subrings. Finally, we construct many fundamental properties which characteristic direct product of finite intuitionistic fuzzy normed normal subrings.