I am having trouble doing the following question (I'm studying for quals, it isn't homework)
If $A$ is a noetherian integral domain such that for every maximal $m\subset A$, the quotient $m/m^2$ is a one-dimensional vector space over the field $A/m$
(a) Prove every nonzero prime ideal is maximal.
(b) Prove $A$ is integrally closed.
There is a hint which says that one should localize at maximal ideals. My problem is that I'm not really sure how to use the $m/m^2$ condition. A solution or hint in the right direction using a minimal amount of commutative algebra would be much appreciated (but clearly a decent amount should be used).