On Fine Rings and Their Relationship with Clean, Nil-Clean and Polynomial Ring Extensions (المجلة الدولية للبحوث العلمية, 2026)
The structural characteristics of fine rings and their connections to polynomial rings, clean rings, and nil-clean rings are examined in this study. One important class that is closely connected to simple rings is fine rings, which are defined via decompositions involving units and nilpotent elements. We examine whether the fine property is preserved under polynomial extensions and show that, in general, polynomial rings fail to retain this property. Furthermore, we analyze the connections between fine, clean, and nil-clean rings, highlighting that while every fine is structurally rigid and linked to simplicity, clean and nil-clean rings exhibit more flexible behavior, especially in the presence of nilpotent elements. In particular, we study rings of the form Fq[x]/(xn), establishing conditions under which they are clean or nil-clean, and explaining why they are not fine. The results provide a clearer structural comparison between these classes and suggest directions for further research in decomposition theory and ring extensions.