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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.2012-05-23
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    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.2012-05-23
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    @variete Does rad here mean the intersection of primes containing the ideal?2012-05-23
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    Dear @rschwieb: Yep, rad means the radical.2012-05-23
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    @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?)2012-05-23
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    Hmm, that the right is contained in the left is straightforward, but I really cannot complete the other direction :) It's been looking to me that some localization argument might kick in, but I don't know how that plays with the grading.2012-05-23
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    @variete In the LHS, have you written $rad(Ann(M/N^\ast)$ or is it $rad((Ann(M/N)^\ast)$?2012-06-05
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    This question is cross-posted on MO, where it has received an answer that looks reasonable to me.2012-06-07

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