Questions: Chain Conditions and Artinian Rings

Question 1

Which of the following rings is Artinian?

• A. The integers Z
• B. The polynomial ring k[x] over a field k
• C. The quotient ring k[x]/(x^3)
• D. The formal power series ring k[[x]]

Answer: (C)

k[x]/(x^3) is a finite-dimensional k-vector space (basis {1, x, x^2}), so it satisfies DCC on ideals. Its only prime ideal is (x)/(x^3), which is also maximal, consistent with Krull dimension zero. The other rings all have infinite descending chains or infinite Krull dimension.

Question 2

A commutative Artinian ring has Krull dimension zero.

• T. True
• F. False

Answer: True

In an Artinian ring, every prime ideal is maximal. Proof: if P is prime, then R/P is an Artinian domain. In an Artinian domain, every nonzero element is a unit (given 0 ≠ a, the chain (a) ⊇ (a^2) ⊇ ... stabilizes, so a^n = ba^(n+1), giving 1 = ba by cancellation). So R/P is a field, making P maximal. Since every prime is maximal, no chain of primes has length > 0.

Question 3

The ring Z is Noetherian but not Artinian. Give a descending chain of ideals in Z that does not stabilize.

Model answer: (2) ⊃ (4) ⊃ (8) ⊃ (16) ⊃ ... , i.e., (2^n) for n = 1, 2, 3, ..., is a strictly descending chain.

Each (2^n) strictly contains (2^(n+1)) since 2^n is not a multiple of 2^(n+1). This chain never stabilizes, so Z fails DCC. More generally, any Noetherian domain that is not a field fails to be Artinian, since it has Krull dimension at least 1.

Question 4

Every commutative Artinian ring is Noetherian, but the converse is false.

• T. True
• F. False

Answer: True

The Hopkins-Levitzki theorem proves that DCC implies ACC for commutative rings (and more generally for modules over left-Artinian rings). The converse fails because Z is Noetherian but not Artinian. The key asymmetry is that DCC forces Krull dimension 0 and finite length, while ACC allows arbitrary Krull dimension.