Continuous map from $\text{Spec}(R/I)$ to $\text{Spec}(R)$ induced by the canonical projection homomorphism

Question

I want to prove the following:

Let $I$ be an ideal in the ring $R$. Prove that the continuous map from $\text{Spec}(R/I)$ to $\text{Spec}(R)$ induced by the canonical projection homomorphism $R\to R/I$ maps $\text{Spec}(R/I)$ homeomorphically onto the closed set $\mathcal{Z}(I)$ in $\text{Spec}(R)$.

This feels intuitively true, but I am unsure of what direction to take in this proof.

Answer 1

Hint: the result follows easily from the correspondence theorem for ideals of the quotient (this correspondence preserves prime ideals) and the definition of the topology on the spectrum.