What is the Krull dimension of the polynomial ring k[x, y, z] over a field k?
A0
B1
C2
D3
The chain (0) ⊊ (x) ⊊ (x, y) ⊊ (x, y, z) is a chain of prime ideals of length 3 in k[x, y, z]. By the dimension theorem for polynomial rings, dim k[x_1, ..., x_n] = n. No longer chain exists because k[x, y, z] has n = 3 variables.
Question 2 True / False
Krull's Hauptidealsatz states that in a Noetherian ring, every minimal prime over a principal ideal (f) (with f a non-unit, non-zero-divisor) has height exactly 1.
TTrue
FFalse
Answer: False
The Hauptidealsatz says the height is AT MOST 1, not exactly 1. It is possible for f to generate a minimal prime of height 0 if f is a zero divisor. For example, in k[x,y]/(xy), the element x generates (x) whose minimal prime (the image of (x)) has height 0. The theorem provides an upper bound on height, not an exact value.
Question 3 Short Answer
What is the height of the prime ideal (x, y) in k[x, y, z]?
Think about your answer, then reveal below.
Model answer: 2, witnessed by the chain (0) ⊊ (x) ⊊ (x, y).
The height of a prime P is the supremum of lengths of chains of primes descending from P. The chain (0) ⊊ (x) ⊊ (x, y) has length 2. By the generalized Hauptidealsatz (Krull's height theorem), since (x, y) is generated by 2 elements, its height is at most 2. So the height is exactly 2.
Question 4 True / False
A field has Krull dimension 0, and any Artinian ring has Krull dimension 0.
TTrue
FFalse
Answer: True
A field has only one prime ideal (the zero ideal), so the longest chain of primes has length 0. In an Artinian ring, every prime is maximal, so there are no proper containments among primes -- every chain has length 0. Conversely, a Noetherian ring of Krull dimension 0 is Artinian.
Question 5 Short Answer
State the generalized principal ideal theorem (Krull's height theorem) and explain its geometric significance.
Think about your answer, then reveal below.
Model answer: If I = (f_1, ..., f_r) is an ideal generated by r elements in a Noetherian ring, then every minimal prime over I has height at most r. Geometrically: the vanishing locus of r equations has codimension at most r.
This generalizes the Hauptidealsatz from 1 generator to r generators. Geometrically, each equation 'cuts' the ambient space, reducing dimension by at most 1. So r equations produce a variety of codimension at most r. When equality holds (height exactly r), the ideal is said to define a complete intersection.