On flatly generated proper classes of modules

Abstract

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.