Proposition 9

Let I be a right directed preordered set, $(S_\alpha)_{\alpha \in I}$ an increasing family of multiplicative subsets of a ring A and $S = \bigcup_{\alpha \in I} S_\alpha$. We write $\rho_{\beta \alpha} = i^{S_\beta, S_\alpha}_{A^{S_\beta, S_\alpha}}$ for $\alpha \leq \beta$, $\rho_\alpha = i^{S_\alpha, S_\alpha}$. Then $(S_\alpha^{-1}A, \rho_{\beta \alpha})$ is a direct system of Zings and, if, for all $\alpha \in I$, $\rho'_\alpha$ is the canonical mapping of $S_\alpha^{-1}A$ to $\lim \rightarrow S_\alpha^{-1}A$, there exists a unique isomorphism $j$ from $\lim \rightarrow S_\alpha^{-1}A$ to $S^{-1}A$ such that $j \circ \rho'_\alpha = \rho_\alpha$ for all $\alpha \in I$.

For $\alpha \leq \beta \leq \gamma$, $\rho_{\gamma \alpha} = \rho_{\gamma \beta} \circ \rho_{\beta \alpha}$ (no. 1, Corollary 3 to Proposition 2), hence $(S_\alpha^{-1}A, \rho_{\beta \alpha})$ is a direct system. We write $A' = \lim \rightarrow S_\alpha^{-1}A$; as $\rho_\alpha = \rho_\beta \circ \rho_{\beta \alpha}$ for $\alpha \leq \beta$ (no. 1, Corollary 3 to Proposition 2), $(\rho_\alpha)$ is a direct system of homomorphisms and $j = \lim \rightarrow \rho_\alpha$ is the unique homomorphism from $A'$ to $S^{-1}A$ such that $j \circ \rho'_\alpha = \rho_\alpha$ for all $a \in I$. The homomorphisms $\rho'_\alpha \circ i^{S}_{A^\alpha} : A \to A'$ are all equal, for $\rho_{\beta \alpha} \circ i^{S}_{A^\alpha} = i^{S}_{A^\beta}$ for $a \leq \beta$; let $u$ be their common value. Clearly the elements of $u(S)$ are invertible in $A'$, which shows that there exists a homomorphism $h : S^{-1}A \to A'$ such that $h \circ i^{S}_{A} = u$ (no. 1, Proposition 1). Then

for all $\alpha \in I$ and consequently $j \circ h$ is the identity automorphism of $S^{-1}A$. On the other hand, for all $\alpha \in I$,

whence $h \circ j \circ \rho'_\alpha = \rho'_\alpha$ for all $\alpha \in I$; it follows that $h \circ j$ is the identity automorphism of $A'$ and consequently $j$ is an isomorphism.