Applications of Fuzzy Semiprimary Ideals under Group Action

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy $\mathfrak{G}$-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy $\mathfrak{G}$-semiprimary ideal of a ring $\mathcal{R}$ under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under $\mathfrak{G}$-homomorphism of fuzzy ideals.