Let $A$ and $B$ be Noetherian rings and $f: A \rightarrow C$ and $g: B \rightarrow C$ ring homomorphisms. If both $f$ and $g$ are surjective show $\{(a,b) \in A \times B: f(a)=g(b)\}$ is a Noetherian ring.
Here's a hint: show first (I already did this) that if $I_{1},..I_{n}$ are ideals of a ring $A$ such that $I_{1} \cap ... \cap I_{n} = \{0\}$ if each $A/I_{i}$ is Noetherian then so is $A$.
How to apply the above result?