Noetherianess of a locally noetherian affine scheme without axiom of choice

Question

I use the definition of a noetherian ring given by Qiaochu in this: A commutative ring is noetherian if, for any nonempty collection of ideals $\mathcal{I}$, there is some $I \in \mathcal{I}$ which is not properly contained in any $J \in \mathcal{I}$.

Can we prove the following theorem without using Axiom of Choice?

Theorem Let $A$ be a commutative ring with unity. Let $f_1,\cdots, f_n$ be elements of $A$ such that $A = (f_1,\cdots, f_n)$. Suppose that each $A_{f_i}$ is noetherian. Then $A$ is noetherian.

As for why I think this question is interesting, please see(particularly Pete Clark's answer): Why worry about the axiom of choice?

Remark Under the condition of the theorem, we can prove without the axiom of choice that $A$ satisfies the ascending chain condition(see for example Hartshorne). However, we need the axiom of dependent choice to prove that $A$ is noetherian in our sense using this fact.

Answer 1

Notation Let $A$ be a commutative ring with unity. Let $I$ be an ideal of $A$. Let $f$ be an element of $A$. We denote by $I_{(f)}$ the inverse image of $IA_f$ by the canonical homomorphism $A \rightarrow A_f$. Namely $I_{(f)} = \{x \in A | \ $ There exists an integer $n \gt 0$ such that $f^n x \in I \}$.

Let $\mathcal{I}$ be a nonempty collection of ideals of $A$. We denote by $\mathcal{I}_{(f)}$ the set $\{I_{(f)}\ | \ I \in \mathcal{I} \}$.

Lemma 1 Let $A$ be a commutative ring with unity. Let $f$ be an element of $A$. Let $\mathcal{I}$ be a nonempty collection of ideals of $A$. Let $J$ be an element of $\mathcal{I}$. Suppose $JA_f$ is maximal in the set $\{IA_f \ |\ I \in \mathcal{I} \}$. Then $J_{(f)}$ is maximal in $\mathcal{I}_{(f)}$.

Proof: Let $I$ be an element of $\mathcal{I}$ such that $J_{(f)} \subset I_{(f)}$. Then $J_{(f)}A_f \subset I_{(f)}A_f$. Hence $JA_f \subset IA_f$. By the assumption, $JA_f = IA_f$. Hence $J_{(f)} = I_{(f)}$. QED

Lemma 2 Let $A$ be a commutative ring with unity. Let $f_1,\cdots, f_n$ be elements of $A$ such that $A = (f_1,\cdots, f_n)$. Let $I, J$ be ideals of $A$ such that $I \subset J$. Suppose $I_{(f_i)} = J_{(f_i)}$ for all $i$. Then $I = J$.

Proof: Let $x \in J$. It suffices to prove that $x \in I$. Since $J \subset J_{(f_i)}$ for all $i$, $J \subset I_{(f_i)}$ for all $i$, Hence there exists an integer $m \gt 0$ such that $f_i^m x \in I$ for all $i$. By this question, we can prove without the axiom of choice that $A = (f_1^m,\cdots, f_n^m)$. Hence there exists $g_1,\cdots, g_n \in A$ such that $1 = g_1f_1^m + \cdots + g_n f_n^m$. Then $x = g_1f_1^m x + \cdots + g_n f_n^m x \in I$ as desired. QED

Proof of the theorem We define inductively sets of nonempty ideals $\mathcal{I}_0, \mathcal{I}_1, \cdots, \mathcal{I}_n$ and ideals $J_1, \cdots, J_n$ as follows. We define $\mathcal{I}_0 = \mathcal{I}$. Let $k$ be an integer such that $1 \le k \le n$. Suppose $\mathcal{I}_{k-1}$ is defined. Let $J_k$ be an element of $\mathcal{I}_{k-1}$ such that ${J_k}_{(f_k)}$ is a maximal element of $\mathcal{I_{k-1}}_{(f_k)}$ which exists by Lemma 1. We define $\mathcal{I}_k = \{I \in \mathcal{I}_{k-1} \ |\ I_{(f_k)} = {J_k}_{(f_k)}\}$. We claim that $J_n$ is a maximal element of $\mathcal{I}$. Let $I \in \mathcal{I}$ be such that $J_n \subset I$.

We first show by induction that $I \in \mathcal{I}_k$ for $k = 0, 1, \cdots, n$. Since $\mathcal{I}_0 = \mathcal{I}$, $I \in \mathcal{I}_0$. Let $k$ be an integer such that $1 \le k \le n$. Suppose $I \in \mathcal{I}_{k-1}$. Since $J_n \in \mathcal{I}_k$, ${J_n}_{(f_k)} = {J_k}_{(f_k)}$. Since ${J_n}_{(f_k)} \subset I_{(f_k)}$, ${J_k}_{(f_k)} \subset I_{(f_k)}$. Since $I \in \mathcal{I}_{k-1}$, ${J_k}_{(f_k)} = I_{(f_k)}$. Hence $I \in \mathcal{I}_k$ as desired.

Next we prove that ${J_n}_{(f_k)} = I_{(f_k)}$ for all $k$. Since $J_n \in \mathcal{I}_k$, ${J_n}_{(f_k)} = {J_k}_{(f_k)}$. On the other hand, since $I \in \mathcal{I}_k$, ${J_k}_{(f_k)} = I_{(f_k)}$. Hence ${J_n}_{(f_k)} = I_{(f_k)}$ as desired.

By Lemma 2, $J_n = I$. Hence $J_n$ is a maximal element of $\mathcal{I}$. QED