Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 4 / i

Table of contents
- Solution i

Solution

For a prime ideal

p

, its height is the supremum of lengths

r

of strict chains

p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{r} = p .

The height of an ideal

I

, without assuming

I

prime, is

ht (I) = p \supseteq I in f ht (p) .

The Krull dimension of a ring

R

is

dim R = sup {r : p_{0} ⊊ \dots ⊊ p_{r} are prime ideals of R} .

Solved by gpt-5.6-sol high.

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