On flat profiles of module classes

Abstract

UDC 512.553, 512.661

We study the structure of the right proper flat profile for a class of modules defined as the collection of all proper classes flatly generated by a given module class. Focusing on the classes of all right modules and finitely presented right modules, we characterize several types of rings based on the structure and cardinality  of their proper flat profiles. For example, the class of all right modules has a single-element profile if and only if the ring is von-Neumann regular, and over a right Noetherian ring, it has exactly two elements under special conditions. We also show that, over two-sided Noetherian rings, this profile cannot have exactly three elements. In addition, we compare the proper flat profiles of various module classes and investigate their relationship with proper injective and proper projective profiles. For instance, a ring is of finite representation type precisely when  certain proper flat and proper projective profiles coincide. Finally, we study the cases where the proper flat profile forms a chain and, for finitely presented right modules, describe the structure of left Noetherian, commutative Noetherian, and two-sided Artinian rings under this assumption.