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

Statement

Let R be a nonzero commutative ring and let I⊊R be a proper ideal. Then

dim⁡(R/I)≤dim⁡R.

Facts & Assumptions

Given: A nonzero commutative ring R and a proper ideal I⊊R.

[L1]

Prime ideals of R/I correspond to prime ideals of R containing I, with strict inclusions preserved (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

[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 R/I lifts to a strict prime chain in R of the same length.

L1given
2.1

Therefore [L2] gives dim⁡(R/I)≤dim⁡R.

L2step 1.1
3.1

So passing to a quotient never increases Krull dimension.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources