# Noetherian Rings in Constructive Mathematics

These definitions are largely from pages 92-93 of Ingo's notes. All rings are commutative with 1. I'm interested in understanding the extent to which discussion of Noetherian rings can be carried over into constructive settings. Consider a standard definition of a Noetherian ring:

A ring $R$ is Noetherian if for every ascending chain of ideals stabilizes; given $I_1 \subseteq I_2 \subseteq \dotsi$, there exists $n$ so that $I_n = I_m$ for all $m \ge n$.

This definition is constructively untenable; the claim that $\mathbb Z$ (or even $\mathbb Z / n \mathbb Z$) is Noetherian implies LPO. This definition will also run into problems if dependent choice fails. We can solve the issue with dependent choice by introducing ascending processes.

Let $M = (M,\preceq)$ be a poset. An ascending process $(S_n)$ with values in $M$ is a sequence $S_0,S_1,S_2,\dotsc$ of inhabited subsets of $M$, such that for each $x \in S_i$, there exists $y \in S_{i+1}$ so that $x \preceq y$. $(S_n)$ is said to halt if $S_n \cap S_{n+1}$ is inhabited for some $n$. We say that $M$ satisfies the ascending process condition of every ascending process with values in $M$ halts. We say that $M$ satisfies the weak ascending chain condition if for every every ascending chain $(x_i)$, we have $x_{i+1} \preceq x_i$ for some $i$.

We say that $R$ is Process-Noetherian if the poset of finitely generated ideals in $R$ satisfies the ascending process condition.

Let $R$ be a ring. Let $M \subseteq \mathcal P(R)$ be the poset of ideals, and let $M_F \subseteq M$ be the poset of finitely generated ideals. Assuming dependent choice and excluded middle, the following candidate definitions of Noetherianity are easily shown to be equivalent:

1. $M = M_F$
2. $M$ satisfies the weak ascending chain condition
3. $M_F$ satisfies the weak ascending chain condition
4. $M$ satisfies the ascending process condition
5. $M_F$ satisfies the ascending process condition ($R$ is process-Noetherian)
6. For every sequence $(x_i)_{i \ge 1}$ of elements of $R$, there exists $n \ge 0$ so that $x_{n+1} \in (x_1,\dotsc,x_n)$.

If excluded middle fails, then properties $1,2,4$ all fail when $R = \mathbb Z$, and so are completely hopeless. We have $5 \Rightarrow 3 \Rightarrow 6$ unconditionally, and it doesn't seem reasonable to expect that $3 \Rightarrow 5$ without dependent choice, and so $3$ and $6$ seem unlikely work in the absence of dependent choice.

Question 1: Is the implication $3 \Rightarrow 6$ reversible? What if we assume dependent choice? Question 2: Can we constructively prove the analogue of Hilbert's basis theorem using any of the three not-completely-hopeless definitions? What changes if we assume dependent choice?

• I don't have time to look at the question right now, but will do so later this day or tomorrow. Meanwhile, in addition to the very useful resources cited by Andrej, you can also have a look at the answers (and the references listed therein) of mathoverflow.net/questions/222923/…. – Ingo Blechschmidt May 24 '16 at 7:58