On monoid graded semihereditary rings

TL;DR

This paper develops graded module theory for monoid-graded rings, providing characterizations of graded semihereditary and hereditary rings, including graded Pr"ufer and Dedekind domains.

Contribution

It introduces graded versions of key module concepts and criteria, enabling new characterizations of graded semihereditary and hereditary rings.

Findings

Provides graded Baer's criterion for injectivity.

Establishes graded Lazard's theorem on flatness.

Characterizes graded-Pr"ufer and graded-Dedekind domains.

Abstract

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-Pr\"{u}fer and graded-Dedekind domains.