Show $\varphi:R/I\to R/J$ is a well-defined ring homomorphism

Question

Let $I\subset J\subset R$ be a chain of ideals and define a function $\varphi:R/I\to R/J$ by $\varphi(r+I)=r+J$. Then, I would like to prove that $\varphi$ is a well-defined ring homomorphism. To show that $\varphi$ is a ring homomorphism, I think I need to show that for any $a+I,b+I\in R/I$,

(1) $\varphi((a+I)+(b+I))=\varphi(a+I)+\varphi(b+I)$

(2) $\varphi((a+I)(b+I))=\varphi(a+I)\varphi(b+I)$.

Is this correct?

I also do not quite understand how to prove well-definedness. I understand the general concept- that no matter what representative is used to represent a given coset, $\varphi$ still maps this coset to the same value in $R/J$. I think I have to show this before I show that $\varphi$ is a ring homomorphism, but I am unsure as to what information I can use if I don't know anything about $\varphi$.

Answer 1

Well-definedness: For $r+I=r'+I$, we need to show that $r+J=r'+J$. For suppose $r+I=r'+I$, then $r-r'\in I\subseteq J$, so $r-r'\in J$, and hence $r+J=r'+J$.

Answer 2

It is well-defined if you can show that $$a+I=a'+I\implies a+J=a'+J.$$ This is clear: the hypothesis implies $$(a+I)+J=a+(I+J)=(a'+I)+J=a'+(I+J),$$ and you just have to observe that, since $I\subset J$, $\; I+J=J$.