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**1. Homework Statement**

[itex]R,M[/itex] are Noetherian. Prove that the radical of the annihilator of an [itex]R[/itex]-module[itex]M[/itex], [itex]Rad(ann(M))[/itex]

is equal to the intersection of the prime ideals in the set of associated primes of [itex]M[/itex] (that is denoted so regretfully that I am not even allowed to spell it out by the system)

**2. Homework Equations**

This result is proved in exercises in Dummit and Foote by using tools like the support of [itex]M[/itex] and Zarski topology. In Hungerford, it appears early, just after definition of primary submodules, hence there should be a simple solution that I do not see.

Associated primes are such annihilators of elements of the module that the ideal is actually prime. Question comes up in the exercise section after introducing the notion of primary decomposition, without notion of ring spectrum, Zarski topology, module localizations and such.

**3. The Attempt at a Solution**

I am not looking for a solution, but just for a tip as where to start. It seems like the exercise should not be hard at all, but I've had some problems. One way inclusion is easy. Since annihilator of [itex]M[/itex] is the intersection of annihilators of all elements of [itex]M[/itex], annihilator of [itex]M[/itex] certainly is contained in every annihilator of an element, hence every associated prime. Radical of annihilator of [itex]M[/itex] is the set of prime ideals containing it, so it is at most as big as the intersection of associated primes.

And here I have my problem. I try a proof by contradiction, working with an element of the intersection of associated primes that is not in [itex]Rad(ann(M))[/itex]. Because of the characterization of the radical of an ideal as the intersection of prime ideals containing it, if the chosen element is not in the radical, then there must exist a prime ideal that contains the radical, but not the element itself. I'm thinking of deriving a contradiction there, could this be a way to go? For example, the section introduced primary decomposition, and annihilator of M has such a decomposition. How would it look?

Could anyone point me, gently, in the right direction?