Map to localization is injective on spectra

This article gives the statement, and possibly proof, of a fact about how a property of a homomorphism of commutative unital rings, forces a property for the induced map on spectra
View other facts about induced maps on spectra

Statement

Set-theoretic statement

Suppose $R$ is a commutative unital ring, $U$ is a multiplicatively closed subset of $R$ and $S=U^{-1}R$ is the localization of $R$ at the multiplicatively closed subset $U$. Then the induced map on spectra:

$Spec(S)\to Spec(R)$

is injective. In fact:

• The image of this map is those primes $P$ that are disjoint from $U$
• The inverse image of a prime ideal $P$ is precisely the prime ideal $U^{-1}P$ i.e. the extension of $P$ to $S$

Topological statement

Further, it is also true that the topology on $Spec(S)$ is such that if we give the subspace topology to its image in $Spec(R)$, the map is a homeomorphism.