Proof that $(A/I)/(J/I)$ is isomorphic to $A/J$

Question

Let $I \subset J$ be ideals in $A$. I want to prove that $(A/I)/(J/I)$ is isomorphic to $A/J$. I am guessing I have to use the first isomorphism theorem to deduce this but I can't find the right map to use this theorem. Can someone help me?

Answer 1

Define

$$\phi: A/I\to A/J\;,\;\;\;\phi(a+I):=a+J$$

Observe the map is well defined since

$$a+I=a'+I\implies a-a'\in I\subset J\implies a+J=s'+J$$

and observe also that

$$a+I\in\ker \phi\;\iff\;a\in J\iff a+I\in J/I$$

Now apply the first isomorphism theorem...