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 $a+I,b+I\in R/I$,
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 $\varphi$.