Editing Commutative ring (section)

=== Dimension ===
{{Main|Krull dimension}}
The ''Krull dimension'' (or dimension) dim ''R'' of a ring ''R'' measures the "size" of a ring by, roughly speaking, counting independent elements in ''R''. The dimension of algebras over a field ''k'' can be axiomatized by four properties:
* The dimension is a local property: {{nowrap|1=dim ''R'' = sup<sub>p ∊ Spec ''R''</sub> dim ''R''<sub>''p''</sub>}}.
* The dimension is independent of nilpotent elements: if {{nowrap|''I'' ⊆ ''R''}} is nilpotent then {{nowrap|1=dim ''R'' = dim ''R'' / ''I''}}.
* The dimension remains constant under a finite extension: if ''S'' is an ''R''-algebra which is finitely generated as an ''R''-module, then dim ''S'' = dim ''R''.
* The dimension is calibrated by dim {{nowrap|1=''k''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>] = ''n''}}. This axiom is motivated by regarding the polynomial ring in ''n'' variables as an algebraic analogue of [[affine space|''n''-dimensional space]].

The dimension is defined, for any ring ''R'', as the supremum of lengths ''n'' of chains of prime ideals
{{block indent|1= ''p''<sub>0</sub> ⊊ ''p''<sub>1</sub> ⊊ ... ⊊ ''p''<sub>''n''</sub>. }}
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ (''p''), where ''p'' is a [[prime number]]. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in [[commutative algebra]], the [[going-up theorem]] and [[Krull's principal ideal theorem]].