Suppose $R$ is a ring and $S$ a multiplicative set in $R$. Then the localization $S^{-1}R$ satisfies the following universal property:
- Every element of $S$ maps to an invertible element in $S^{-1}R$.
- If $R\to T$ is a ring homomorphism such that every element of $S$ maps to an invertible element in $T$, then there is a unique ring homomorphism $S^{-1}R \to T$.
I was wondering whether the following property holds:
If $W$ is a multiplicative subset of $S$ then there is a unique homomorphism (satisfying certain mild conditions) $W^{-1}R \to S^{-1}R$?
I think there is a natural map which sends $r/w \to r/w$, but is it the only one?
Essentially if we invert more elements than $S$ we get a map from the localization. If we invert less elements than $S$ do we get a map to the localization?