Mathematics.

rings

Noetherian Rings

Abstract Algebra II30 minDifficulty7 out of 10

You should know: ideals

Overview

A ring R is Noetherian, named after Emmy Noether, if its ideals cannot form an infinitely increasing chain — every ascending sequence of ideals I₁ ⊆ I₂ ⊆ I₃ ⊆ ... must eventually stabilize. This single finiteness condition, equivalent to every ideal being finitely generated, is what makes most of the rings used in practice (ℤ, polynomial rings over a field, and any of their quotients) tame enough to support induction arguments, the Hilbert basis theorem, and primary decomposition. Noether isolated this property in the 1920s precisely because it is the weakest hypothesis under which 'ideal theory' behaves the way it does for ℤ.

Intuition

Think of building an ideal by throwing in generators one at a time: (a₁) ⊆ (a₁,a₂) ⊆ (a₁,a₂,a₃) ⊆ .... In a Noetherian ring this process must terminate after finitely many steps no matter which elements you choose, because the chain of ideals it produces has to stabilize. Equivalently, no ideal in a Noetherian ring can be so complicated that it needs infinitely many generators — every ideal, however large, is captured by a finite list. This is what lets proofs proceed by picking a maximal counterexample: in a Noetherian ring, if a nonempty collection of 'bad' ideals existed, the maximal condition guarantees one of them is maximal among bad ideals, and that maximal bad ideal can usually be shown to not exist, giving a contradiction.

Formal Definition

Definition

Let R be a commutative ring with 1. The following conditions on R are equivalent, and R is called Noetherian when any one (hence all) hold:

I1⊆I2⊆I3⊆⋯  ⟹  ∃ N:In=IN for all n≥NI_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots \implies \exists\, N: I_n = I_N \ \text{for all } n \geq N
Ascending chain condition (ACC) on ideals
every ideal I⊆R is finitely generated: I=(a1,…,ak) for some ai∈R\text{every ideal } I \subseteq R \text{ is finitely generated: } I = (a_1, \ldots, a_k) \text{ for some } a_i \in R
Finite generation of ideals
every nonempty set of ideals of R has a maximal element (w.r.t. ⊆)\text{every nonempty set of ideals of } R \text{ has a maximal element (w.r.t. } \subseteq \text{)}
Maximal condition

Properties

ACC ⟺ finite generation

R satisfies ACC on ideals  ⟺  every ideal of R is finitely generated.R \text{ satisfies ACC on ideals} \iff \text{every ideal of } R \text{ is finitely generated.}

Hilbert Basis Theorem

R Noetherian  ⟹  R[x] Noetherian (and hence R[x1,…,xn] Noetherian).R \text{ Noetherian} \implies R[x] \text{ Noetherian (and hence } R[x_1,\ldots,x_n] \text{ Noetherian).}

Quotients and images

R Noetherian and I⊴R  ⟹  R/I Noetherian.R \text{ Noetherian and } I \trianglelefteq R \implies R/I \text{ Noetherian.}

PIDs are Noetherian

Every principal ideal domain is Noetherian, since every ideal is generated by a single element.\text{Every principal ideal domain is Noetherian, since every ideal is generated by a single element.}

Fields are Noetherian

A field has only the ideals (0) and the whole field, so ACC holds trivially.\text{A field has only the ideals } (0) \text{ and the whole field, so ACC holds trivially.}

Worked Examples

  1. Every ideal of ℤ is principal: any ideal I ⊆ ℤ equals (n) for the smallest positive element n ∈ I (or I=(0)).

    I=(n), n=min⁡{m∈I:m>0}I = (n),\ n = \min\{ m \in I : m > 0 \}

  2. Given a chain (n₁) ⊆ (n₂) ⊆ ..., containment (n_{k+1}) ⊆ (n_k) forces n_k \mid n_{k+1}... but ascending containment (n_k) ⊆ (n_{k+1}) means n_{k+1} \mid n_k, so |n_{k+1}| \le |n_k| for each step.

    (nk)⊆(nk+1)  ⟹  nk+1∣nk  ⟹  ∣nk+1∣≤∣nk∣(n_k) \subseteq (n_{k+1}) \implies n_{k+1} \mid n_k \implies |n_{k+1}| \le |n_k|

  3. A strictly decreasing sequence of positive integers |n_1| \ge |n_2| \ge \cdots cannot continue forever, so the chain of divisors — and hence the chain of ideals — must stabilize.

    ∣n1∣≥∣n2∣≥⋯ stabilizes after finitely many steps|n_1| \ge |n_2| \ge \cdots \ \text{stabilizes after finitely many steps}

Answer: Every ideal of ℤ is principal, and any ascending chain of principal ideals corresponds to a non-increasing sequence of positive generators, which must stabilize — so ℤ satisfies ACC and is Noetherian.

Practice Problems

Difficulty 6/10

Using the Hilbert Basis Theorem, explain why ℤ[x, y] is Noetherian.

Difficulty 7/10

Is every subring of a Noetherian ring Noetherian? Give a reason or a counterexample.

Difficulty 8/10

Why does the ascending chain condition guarantee that every ideal in a Noetherian ring is finitely generated? Sketch the argument.

Common Mistakes

Common Mistake

Believing 'Noetherian' means 'has finitely many ideals'.

Noetherian rings can have infinitely many ideals (ℤ has infinitely many ideals (n)); the requirement is only that ascending chains stabilize, not that the total collection of ideals is finite.

Common Mistake

Assuming every subring of a Noetherian ring is Noetherian.

Noetherian-ness is inherited by quotients and by polynomial extensions (Hilbert Basis Theorem), but not automatically by arbitrary subrings.

Quiz

A ring R is Noetherian precisely when:
The Hilbert Basis Theorem states that if R is Noetherian, then:
Which of these rings is NOT Noetherian?

Summary

  • R is Noetherian iff every ascending chain of ideals stabilizes (ACC), equivalently iff every ideal is finitely generated.
  • PIDs (like ℤ) and fields are automatically Noetherian, since their ideals are trivially finitely generated.
  • The Hilbert Basis Theorem: R Noetherian ⟹ R[x] Noetherian, extending to any finite number of variables by induction.
  • Infinitely-many-variable polynomial rings are the standard counterexample: the ideal of all variables has no finite generating set.
  • ACC underlies maximal-element proof techniques throughout ideal theory, including primary decomposition and the theory of Noetherian modules.

References

  1. WebsiteWikipedia — Noetherian ring
  2. WebsiteWikipedia — Hilbert's basis theorem