A characterization of monoid graded semihereditary rings

Let $\mathrm{\Gamma }$ be a cancellation monoid and $R={⨁}_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ be a $\mathrm{\Gamma }$-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-Prüfer domains.