Let $G$ be an abelian group and $R=\oplus_{g\in G}R_g$ be a $G$-graded, commutative ring with unity . Let us call $R$ to be a "Graded PID" if every graded ideal of $R$ is generated by a homogeneous element . Let $M=\oplus_{g\in G}M_g$ be a $G$-graded, free module over a graded PID, $R=\oplus_{g\in G}R_g$ , then is it true that $M$ is graded-free ? i.e. does $M$ has a basis over $R$ consisting of homogeneous elements ? If this is difficult to answer in general , then what if we also assume that $M$ is finitely generated , or if we assume $G=\mathbb Z$ ?
( Note that $M$ is graded and free implies $M$ is graded and projective or equivalently that $M$ is graded-projective )