On monoid graded semihereditary rings

In order to study graded left hereditary and left semihereditary rings graded by a cancelation monoid in terms of their modules, we need to revisit graded free, projective, injective, and flat modules and provide graded versions of specific results concerning these modules, like Baer’s criterion on injectivity and Lazard’s theorem on flatness. Then, among other things, we can give some characterization of graded left hereditary and left semihereditary rings, in particular, of graded-Prüfer and graded-Dedekind domains.