integral domain as an intersection of localizations

Question

I am stuck on trying to prove that $$\bigcap_{m \text{ maximal}}A_m = A$$ where $A$ is an integral domain (commutative, ring with unity). I would appreciate any hint! Thanks!

Answer 1

Hint $\ $ If fraction $\rm\:f\not\in A\:$ then its denominator ideal $\rm\: I = \{ d\in A \mid d\:\!f\in A\} \ne (1),\:$ so $\rm\!\: I\!\:$ is contained in a maximal ideal $\rm\:M,\:$ so $\rm\:f\not\in A_M,\:$ thus $\rm\:f\:$ is not in the intersection. The converse is clear.