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

Question

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.

Answer 1

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.

Answer 2

A homomorphism $\phi\colon R\to S$ induces a homomorphism $R/I\to S$ iff $I\subseteq \ker \phi$.