It is well known that the associated primes of a module over a commutative ring (those primes associated to primary decomposition of the zero submodule, provided such decomposition exists) are precisely the radicals of the annihilators of elements of the module that are prime. It is easy to show that the word "radical" can be omitted if the ring is Noetherian. Apparently it can also be omitted if the module (and not the ring) is Noetherian. The only proof I know constructs a huge theory of injective modules, and I'm curious to know if there is a more elementary proof.
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