---
tag: GFVWEATKGE
kind: concept
name: Dimension theory
msc: [13C15]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_theory_(algebra)
same_as:
  wikidata: Q5277265
  wikipedia: Dimension theory (algebra)
---

## Summary

Dimension theory is the part of commutative algebra that studies the [[Krull dimension|MVGBS3J1P7]] of rings and modules.
It also studies related notions such as [[height|87WH65FJB8]], [[system of parameters|95QTEPMDA4]] and [[multiplicity|JX20MNDDXT]].
Its central results are [[Krull's principal ideal theorem|SJZW2HN1SC]] and the [[dimension theorem for local rings|KW3HY77V6D]].
The latter computes the Krull dimension of a Noetherian local ring from its [[Hilbert–Samuel function|BNQVS83GZV]].
It gives an algebraic meaning to the dimension of an algebraic variety.

## Explanations

### L1 human

A line has dimension $1$ and a plane has dimension $2$.
Dimension theory explains what dimension means for an arbitrary commutative ring, where there is no picture to look at.
The main idea is to count the longest chain of prime ideals, and then to show that this count agrees with other natural counts.
For example, the polynomial ring $k[x, y]$ over a field has the chain $0 \subsetneq (x) \subsetneq (x, y)$, and its dimension is $2$, as for the plane.