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 tau-rugged, as opposed to flat modules. Properties of tau-rugged modules are studied. We study the structure of rings whose certain types of modules are either flat or tau-rugged. Specifically, we prove that if R is a right Noetherian ring, then every (finitely presented) nonflat right R-module is tau-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module and it is either Artinian serial ring with J(2)(R) = 0 or right finitely & sum;-CS, right SI ring. If R is a right perfect ring with nonflat finitely presented simple right R-module U, then U is tau-rugged if and only if every nonflat right R-module is tau-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module U and it is Artinian serial ring with J(2)(R) = 0. In addition, if R is commutative and every nonflat R-module is tau-rugged, then R is either a von Neumann regular ring or an fp-injective ring.