Exercise 1, Appendix 0

Let $I$ be a nonempty right filtered preordered set and let $(A_\alpha,\ \varphi_{\beta\alpha})$ be an inductive system of rings relative to $I$. For each index $\alpha$ in $I$, let an ideal $q_\alpha$ of $A_\alpha$ be given. Assume that one has $\varphi_{\beta\alpha}(q_\alpha)\ A_\beta = q_\beta$ for $\beta \geqslant \alpha$. Let $A$ denote the inductive limit of the $A_\alpha$, and for $\alpha \in I$, let $\varphi_\alpha : A_\alpha \to A$ denote the canonical homomorphism. Put

a) If for every $\alpha \in I$, $q_\alpha$ is a maximal ideal of $A_\alpha$, then $q$ is a maximal ideal of $A$. b) If for every $\alpha \in I$, $q_\alpha$ is contained in the radical of $A_\alpha$, then $q$ is contained in the radical of $A$. c) If for every $\alpha \in I$ and every $\beta \in I$, such that $\beta \geqslant \alpha$, the homomorphism $\varphi_{\beta\alpha}$ is faithfully flat, then for every $\alpha \in I$, $\varphi_\alpha$ is faithfully flat. We shall henceforth assume that the hypothesis of c) is satisfied. d) If for every $\alpha \in I$, $A_\alpha$ is separated for the $q_\alpha$-adic topology, then $A$ is separated for the $q$-adic topology. e) If for every $\alpha \in I$, $A_\alpha$ is noetherian and if $A/q$ is noetherian, then $(1 + q)^{-1}A$ is noetherian. (Reason as in the proof of Prop. 1 of the Appendix, using in addition the following result, which will be proved: Let $B$ be a commutative ring, $p$ an ideal of $B$, $\hat{B}$ the separated completion of $B$ for the $p$-adic topology, and $i : B \to \hat{B}$ the canonical mapping. Then there exists one and only one ring homomorphism $j : (1 + p)^{-1}B \to \hat{B}$ extending $i$. For every maximal ideal $m$ of $(1 + p)^{-1}B$, one has $j(m) \hat{B} \neq \hat{B}$.)