A characterization of monoid graded semihereditary rings

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.