TITLE: Annihilator of a maximal ideal in a ring
QUESTION [5 upvotes]: Let R be a ring and M a maximal ideal in R. Prove or disprove: If M is contained in the set of zero divisors of R, then ann(M) is not 0.   

It is easy to see that the statement is true when M is a principal ideal.
But I don't know how to solve for the general case.

REPLY [2 votes]: As Censi Li already showed (+1), that statement is wrong for a general commutative ring. However, adding the assumption that $R$ is Noetherian, it is true and follows from the Prime Avoidance Lemma together with the fact that the set of zero-divisors is the union of the set of Associated Primes