Editing Commutative ring (section)

=== Complete intersections ===
[[File:Twisted_cubic_curve.png|thumb|The [[twisted cubic]] (green) is a set-theoretic complete intersection, but not a complete intersection.]]
By [[Krull's principal ideal theorem]], a foundational result in the [[dimension theory (algebra)|dimension theory of rings]], the dimension of
{{block indent|1= ''R'' = ''k''[''T''<sub>1</sub>, ..., ''T''<sub>''r''</sub>] / (''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>) }}
is at least ''r'' &minus; ''n''. A ring ''R'' is called a [[complete intersection ring]] if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings. Any regular local ring is a complete intersection ring, but not conversely.

A ring ''R'' is a ''set-theoretic'' complete intersection if the reduced ring associated to ''R'', i.e., the one obtained by dividing out all nilpotent elements, is a complete intersection. As of 2017, it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections.{{sfn|Lyubeznik|1989|ps=}}