Flat-precover completing domains

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