# Radical of annihilator of a module

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 $Ann_{R}M/N)^{*}$

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What have you tried so far? Use the definitions and try to show at least one inclusion. – Martin Brandenburg May 23 '12 at 8:14
which inclusion do you mean ? Thank for reminding me, but I have got no idea to prove it. I just take one element belong to rad$Ann_{R}M/N^{*}$ and try to prove it belong to the RHS, but I do not know to prove. – variete May 23 '12 at 9:09
@variete Does rad here mean the intersection of primes containing the ideal? – rschwieb May 23 '12 at 11:21
Dear @rschwieb: Yep, rad means the radical. – variete May 23 '12 at 11:31
@variete Just checking, because rad is also used to denote the Jacobson radical of a module. (It'd be good to include this in the question.) And the grading is either over Z or N (nothing more complicated?) – rschwieb May 23 '12 at 11:33