Questions: Noetherian Rings

Question 1

Which of the following is NOT equivalent to the statement that a commutative ring R is Noetherian?

• A. Every ideal of R is finitely generated
• B. Every ascending chain I₁ ⊆ I₂ ⊆ I₃ ⊆ ··· of ideals in R eventually stabilizes
• C. Every nonempty set of ideals of R has a maximal element under inclusion
• D. Every prime ideal of R is principal

Answer: (D)

The first three options are the three equivalent formulations of the Noetherian condition: finite generation of all ideals, the ascending chain condition (ACC), and the maximal condition. The fourth option describes principal ideal domains, which are Noetherian but form a much smaller class. The ring ℤ[x] is Noetherian (by the Hilbert basis theorem) but not a PID, since the ideal (2, x) is not principal. Having all primes principal is neither necessary nor sufficient for the Noetherian property in general.

Question 2

A student claims that the polynomial ring k[x₁, x₂, x₃, ...] in infinitely many variables over a field k is Noetherian because k is a field. What is wrong with this argument?

• A. The argument is actually correct — polynomial rings over fields are always Noetherian
• B. The Hilbert basis theorem says R[x] is Noetherian when R is, but it adds one variable at a time; the ring of infinitely many variables is not Noetherian because the chain (x₁) ⊆ (x₁, x₂) ⊆ (x₁, x₂, x₃) ⊆ ··· never stabilizes
• C. The ring is Noetherian but only if k is algebraically closed
• D. k[x₁, x₂, ...] is Noetherian because each individual ideal involves only finitely many variables

Answer: (B)

The Hilbert basis theorem guarantees that k[x₁, ..., xₙ] is Noetherian for every finite n, by induction. But k[x₁, x₂, ...] is the union (direct limit) of these rings, and the ascending chain (x₁) ⊆ (x₁, x₂) ⊆ (x₁, x₂, x₃) ⊆ ··· never stabilizes — at each step the new variable xₙ₊₁ is not in the previous ideal. This violates the ACC, so the ring is not Noetherian. The Noetherian property does not survive passage to infinitely many variables.

Question 3

Every principal ideal domain is a Noetherian ring.

• T. True
• F. False

Answer: True

In a PID, every ideal is generated by a single element — in particular, every ideal is finitely generated (by one element). Since the Noetherian condition requires every ideal to be finitely generated, PIDs automatically satisfy it. Examples include ℤ and k[x] for any field k. The converse fails: ℤ[x] is Noetherian (by the Hilbert basis theorem) but not a PID.

Question 4

Every subring of a Noetherian ring is Noetherian.

• T. True
• F. False

Answer: False

Consider the polynomial ring k[x₁, x₂, ...] in infinitely many variables, which is a subring of its own fraction field (a field, hence Noetherian). But k[x₁, x₂, ...] is not Noetherian. More concretely, ℤ is Noetherian, but subrings of Noetherian rings need not be Noetherian in general. The correct statement is that quotient rings and localizations of Noetherian rings are Noetherian, but subrings need not be.

Question 5

Explain why the ascending chain condition on ideals is equivalent to every ideal being finitely generated, and why this equivalence matters for commutative algebra.

Model answer: Suppose every ideal is finitely generated, and let I₁ ⊆ I₂ ⊆ ··· be an ascending chain. The union I = ∪Iₙ is an ideal, so I = (a₁, ..., aₖ) for some finite set. Each aⱼ lies in some Iₙⱼ, so taking N = max(n₁, ..., nₖ), all generators are in Iₙ, giving I = Iₙ and the chain stabilizes. Conversely, if some ideal I were not finitely generated, pick a₁ ∈ I; then (a₁) ≠ I, so pick a₂ ∈ I \ (a₁); then (a₁, a₂) ≠ I, so continue — producing a strictly ascending chain (a₁) ⊊ (a₁, a₂) ⊊ ···, violating ACC.

This equivalence is foundational because different proofs exploit different formulations. The ACC version is used in inductive arguments (choose a maximal element from a collection of ideals). The finite generation version is used when constructing explicit generators. The maximal condition (every nonempty set of ideals has a maximal element) is the Zorn's lemma formulation, used in existence proofs for prime ideals.