# Does a ring homomorphism $\phi: R \rightarrow S$ give rise to any map $\psi: R/I \rightarrow S/J?$

Let $$R, S$$ be rings.

Suppose $$\phi: R \rightarrow S$$ is a ring homomorphism. Clearly we have a map $$R \rightarrow S/J$$ defined by the canonical map.

However, for any ideal $$I \subset R,$$ can I form a ring homomorphism $$R/I \rightarrow S/J?$$

This feels wrong for some reason as $$I, J$$ are unrelated.

However, it seems like I can just send $$a + I \mapsto \phi(a) + I.$$ I don't see anything wrong with this. $$\phi(a + I) + \phi(b + I) = \phi(a) + I + \phi(b) + I = \phi(a + b) + I,$$ and so on...

I feel silly for asking such a basic question but it feels quite strange that this is possible as, again, there is no relation between these two ideals.

Such a definition would be meaningful (i.e. would lead to a ring homomorphism) in case the ideal $$I, J$$ and $$\phi$$ satisfy the condition $$\phi(I)\subset J$$. Otherwise the value will not be well-defined. Taking a different coset representative will give a different value making it NOT a functiton.
A homomorphism $$\phi\colon R\to S$$ induces a homomorphism $$R/I\to S$$ iff $$I\subseteq \ker \phi$$.