A characterization of monoid graded semihereditary rings

TL;DR

This paper characterizes monoid graded semihereditary rings, linking their properties to graded coherence and flatness of submodules, offering a new perspective on graded Prüfer domains.

Contribution

It provides a novel characterization of graded semihereditary rings through graded coherence and flatness conditions, connecting to graded Prüfer domains.

Findings

R is graded left semihereditary iff it is graded left coherent and submodules of flat modules are flat.

The characterization offers a new understanding of graded Prüfer domains.

The results connect properties of graded rings with module-theoretic conditions.

Abstract

Let $\Gamma$ be a cancellation monoid and $R=\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded ring. It is shown that $R$ is graded left semihereditary if and only if $R$ is graded left coherent and every graded submodule of a flat left $R$-module is flat. Hence it gives a new characterization of graded-Pr\"{u}fer domains.