On flatly generated proper classes of modules

There are different mechanisms in the literature to measure how flat a module can be. We study an alternative perspective on the analysis of the flatness of a module, as we assign to every module a class of short exact sequences of modules, namely flatly generated proper classes. We focus on modules that generate flat proper classes, aiming for them to be as small as possible. We refer to such modules as being Formula: see text-rugged, as opposed to flat modules. Properties of Formula: see text-rugged modules are studied. We study the structure of rings whose certain types of modules are either flat or Formula: see text-rugged. Specifically, we prove that if Formula: see text is a right Noetherian ring, then every (finitely presented) nonflat right Formula: see text-module is Formula: see text-rugged if and only if Formula: see text has a unique (up to isomorphism) singular simple right Formula: see text-module and it is either Artinian serial ring with Formula: see text or right finitely Formula: see text-CS, right SI ring. If Formula: see text is a right perfect ring with nonflat finitely presented simple right Formula: see text-module Formula: see text, then Formula: see text is Formula: see text-rugged if and only if every nonflat right Formula: see text-module is Formula: see text-rugged if and only if Formula: see text has a unique (up to isomorphism) singular simple right Formula: see text-module Formula: see text and it is Artinian serial ring with Formula: see text. In addition, if Formula: see text is commutative and every nonflat Formula: see text-module is Formula: see text-rugged, then Formula: see text is either a von Neumann regular ring or an fp-injective ring.