A Detailed Study of Mathematical Rings in q-Rung Orthopair Fuzzy Framework

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the disciplines of mathematics, theoretical physics and coding theory. The study of ideals is vital to the theory of rings in a wide range of ways. The uncertainties present in the information are addressed well by the q-rung orthopair fuzzy set (q-ROFS). Considering the significance of ring theory and the q-ROFS, this article defines q-rung orthopair fuzzy ideals (q-ROFIs) in conventional rings and investigates its various algebraic features. We introduce the notion of q-rung orthopair fuzzy cosets (q-ROFCs) of a q-ROFI and demonstrate that, under certain binary operations, the collection of all q-ROFCs of a q-ROFI forms a ring. In addition, we provide a q-rung orthopair analog of the fundamental theorem of ring homomorphism. Furthermore, we present the notion of q-rung orthopair fuzzy semi-prime ideals (q-ROFSPIs) and provide a comprehensive explanation of their many algebraic properties. Finally, regular rings were characterized using q-ROFIs.