Relationship between $\mathcal{Spec}(R)$ and $\mathcal{Spec}(R_{\text{red}})$

Question

Let $R$ be a commutative ring. I am wondering if there is any well-known relationship between $\mathcal{Spec}(R)$ and $\mathcal{Spec}(R_{\text{red}})$ that would allow one to conclude

$[\mathcal{Spec}(R), \mathbb Z]\cong [\mathcal{Spec}(R_{\text{red}}), \mathbb Z]$,

where $[X, Y]$ denotes the semiring of continuous maps $f: X \to Y$ under pointwise addition and multiplication.

I am asking because I am reading a theorem in some notes that seems to require such a group isomorphism.

Answer 1

$Spec(R)$ and $Spec(R_{red})$ are homeomorphic via the map induced by the projection $R \to R_{red}$.