I met the following problem when I studied graded ring theory. I have no idea to solve it. Please help me. Thank you very much !
Let $R$ be a commutative $\mathbb{Z}$-graded ring, $M$ is a graded R-module, $N$ is a submodule of $M$. Denote by $N^*$ for the submodule of $M$ generated by all the homogeneous elements contained in $N$. Prove that: rad$(Ann_{R}M/N^{*})$=(rad $Ann_{R}M/N)^{*}$