Annihilator of a maximal ideal in a ring

Question

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.

Answer 1

It's not true. Let $F$ be a field, $A:=F[X_1,X_2,\dots]$ the polynomial ring over $F$ in countably many indeterminates. Then taking $R:=A/(X_1,X_2^2,X_3^3,\dots)$ and $M:=(\bar{X}_1,\bar{X}_2,\bar{X}_3,\dots)$ gives a counterexample.

Answer 2

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