Page 707 of Dummit and FooteI am having trouble understanding the argument to prove the universal property of localization.
Quoting Dummit and Foote page 707, $ \Phi(\frac{x}{1})= \phi(x) $ and this implies uniqueness.
Can someone explain why that this is the case?
Thanks in advance for any replies.
Proving uniqueness of functions (including homomorphisms) fulfilling some specific property is usually done by taking two such, and proving that they are equal.
So say we have two homomorphisms $\Psi_1, \Psi_2:D^{-1}R\to S$ with the property that $\Psi_1\circ\pi = \Psi_2\circ\pi = \psi$. It is clear that on any element of the form $\frac x1\in D^{-1}S$, we have $\Psi_1(x/1) = \Psi_2(x/1)$, because by the property they share, they must both equal $\psi(x)$. At the same time, we must have $\Psi_1(1/d)= \Psi_2(1/d)$ for any $d\in D$, as they must both equal $\psi(d)^{-1}$. Finally, for a general element $\frac xd\in D^{-1}R$, we have $\Psi_1(x/d) = \Psi_2(x/d)$ because $\frac{x}{d} = \frac x1\cdot \frac1d$, and $\Psi_1, \Psi_2$ are both ring homomorphisms. Thus we have $\Psi_1 = \Psi_2$.