Q:

noetherian quotients and a finitely generated ideal

Suppose $A$ is a ring and $a$ and $b$ are ideals such that one of them is finitely generated. Also suppose $A/a$ and $A/b$ are noetherian. Apparently this implies $A/ab$ is notherian. 
I was wondering if someone could give me a hint or point me in the write direction because i am pretty stuck. 

A:

Assume $a$ is finitely generated. One has an exact sequence $0\to a/ab\to A/ab\to A/a\to 0$. $a/ab$ is a finitely generated module over $A/b$ and hence Noetherian. Since $A/a$ is Noetherian, we are done.

