Journal of Algebra and its Applications, 2024 (SCI-Expanded)
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 τ-rugged, as opposed to flat modules. Properties of τ-rugged modules are studied. We study the structure of rings whose certain types of modules are either flat or τ-rugged. Specifically, we prove that if R is a right Noetherian ring, then every (finitely presented) nonflat right R-module is τ-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 J2(R) = 0 or right finitely ∑-CS, right SI ring. If R is a right perfect ring with nonflat finitely presented simple right R-module U, then U is τ-rugged if and only if every nonflat right R-module is τ-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 J2(R) = 0. In addition, if R is commutative and every nonflat R-module is τ-rugged, then R is either a von Neumann regular ring or an fp-injective ring.