Asumming that we have two rings $R$ and $S$ and an onto (surjective) homomorphism $\alpha :R\rightarrow S$.
I am interested in knowing what relationships exists between the ideals of $R$ and the ideals of $S$?
One such relationship I can think off (which I am not sure is true) is:
$\alpha(rR) = \alpha(r)S$
What is the most general way to express the relationships by $\alpha$ in between the ideals of the two rings?
Your relation is correct if the homomorphism is surjective. Lore gennerally (with the same hypothesis), the ideals in $S$ correspond bijectively to the ideals of $R$ which contain $\ker\alpha$.