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CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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Localisation does not increase Krull dimension

Statement

Let R be a nonzero commutative ring and let S⊆R be a multiplicative subset with 0∉S. Then

dim⁡(S−1R)≤dim⁡R.

Facts & Assumptions

Given: A nonzero commutative ring R and a multiplicative subset S⊆R with 0∉S.

[L1]

Prime ideals of S−1R correspond to prime ideals of R disjoint from S, with strict inclusions preserved (Prime ideals of a localization are exactly the primes disjoint from the denominator set).

[L2]

Krull dimension is the supremum of the lengths of strict prime chains (Krull dimension of a nonzero ring).

Proof

technique · direct
1.1

By [L1], every strict prime chain in S−1R contracts to a strict prime chain in R of the same length.

L1given
2.1

Taking suprema and using [L2] gives dim⁡(S−1R)≤dim⁡R.

L2step 1.1
3.1

Hence localization does not increase Krull dimension.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources