Group action on intuitionistic fuzzy ideals of Γ-ring

Group actions serve as a powerful tool for exploring the symmetry and automorphism properties of rings. In this paper, we examine group actions on intuitionistic fuzzy ideals (IFIs) within a Γ-ring $\mathcal{M}$. We introduce the concept of the intrinsic product of IFIs in $\mathcal{M}$ and explore various properties of intuitionistic fuzzy prime ideals under the influence of group actions. Further, we propose the notion of an intuitionistic fuzzy $\mathcal{G}$-prime ideal in $\mathcal{M}$. We demonstrate that for an IFI $A$ of $\mathcal{M}$, the ideal $A^{\mathcal{G}} = \bigcap_{g\in \mathcal{G}}A^{g}$ represents the largest $\mathcal{G}$-invariant IFI contained within $A$. Additionally, we establish that the $\mathcal{G}$-primeness of $A^{\mathcal{G}}$ is uniquely characterized by the $\mathcal{G}$-primeness of $A$. Lastly, we examine the behavior of intuitionistic fuzzy $\mathcal{G}$-prime ideals of $\mathcal{M}$ under a $\mathcal{G}$-homomorphism.