The following statement is well-known:

Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M\_{\mathfrak{p}}$ is a free $A\_{\mathfrak{p}}$-module for all $\mathfrak{p}$.

My question is: do we need the assumption that $A$ is Noetherian? I have a proof (from Matsumura) that doesn't require that assumption, but the fact that other references (e.g. Atiyah, Wikipedia) are including this assumption makes me rather uneasy.