Flat-precover completing domains

Abstract

Recently, many authors have embraced the study of certain properties of modules such as projectivity, injectivity and atness from an alternative point of view. Rather than saying a module has a certain property or not, each module is assigned a relative domain which, somehow, measures to which extent it has this particular property. In this work, we introduce a new and fresh perspective on atness of modules. However, we will first investigate a more general context by introducing domains relative to a precovering class X. We call these domains X-precover completing domains. In particular, when X is the class of at modules, we call them at-precover completing domains. This approach allows us to provide a common frame for a number of classical notions. Moreover, some known results are generalized and some classical rings are characterized in terms of these domains.