Some novel properties of complex intuitionistic fuzzy ideals in classical ring

The complex intuitionistic fuzzy set is an extension of the intuitionistic fuzzy set where the membership and non-membership functions are expressed by a complex numbers. Ring theory is a well-known field of abstract algebra that is used in a broad area of present study in mathematics and computer science. The study of ideals is important in numerous ways in ring theory. Keeping in view the importance of complex intuitionistic fuzzy sets and ring theory, in this paper, we define the notion of complex intuitionistic fuzzy ideals in a classical ring $R$ and investigate its various algebraic properties. We obtain that the intersection of any two complex intuitionistic fuzzy ideals of a classical ring $R$ is again a complex intuitionistic fuzzy ideal of $R$. We also define the notion of a complex intuitionistic fuzzy level set. Furthermore, we define the concept of complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal of a classical ring and prove that the set of all complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal forms a ring under certain binary operations. Finally, we prove a complex intuitionistic fuzzy version of the fundamental theorem of a ring homomorphism.