Editing Commutative ring (section)

== Local rings ==
A ring is called [[local ring|local]] if it has only a single maximal ideal, denoted by ''m''. For any (not necessarily local) ring ''R'', the localization
{{block indent|1= ''R''<sub>''p''</sub> }}
at a prime ideal ''p'' is local. This localization reflects the geometric properties of Spec ''R'' "around ''p''". Several notions and problems in commutative algebra can be reduced to the case when ''R'' is local, making local rings a particularly deeply studied class of rings. The [[residue field]] of ''R'' is defined as
{{block indent|1= ''k'' = ''R'' / ''m''. }}
Any ''R''-module ''M'' yields a ''k''-vector space given by {{nowrap|''M'' / ''mM''}}. [[Nakayama's lemma]] shows this passage is preserving important information: a finitely generated module ''M'' is zero if and only if {{nowrap|''M'' / ''mM''}} is zero.

=== Regular local rings ===
[[File:Node_(algebraic_geometry).png|thumb|left|The [[cubic plane curve]] (red) defined by the equation ''y''<sup>2</sup> = ''x''<sup>2</sup>(''x'' + ''1'') is [[singularity (mathematics)|singular]] at the origin, i.e., the ring ''k''[''x'', ''y''] / ''y''<sup>2</sup> &minus; ''x''<sup>2</sup>(''x'' + ''1''), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity.]]
The ''k''-vector space ''m''/''m''<sup>2</sup> is an algebraic incarnation of the [[cotangent space]]. Informally, the elements of ''m'' can be thought of as functions which vanish at the point ''p'', whereas ''m''<sup>2</sup> contains the ones which vanish with order at least 2. For any Noetherian local ring ''R'', the inequality
{{block indent|1= dim<sub>''k''</sub> ''m''/''m''<sup>2</sup> &ge; dim ''R'' }}
holds true, reflecting the idea that the cotangent (or equivalently the tangent) space has at least the dimension of the space Spec ''R''. If equality holds true in this estimate, ''R'' is called a [[regular local ring]]. A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the [[tangent cone]])
<math display="block">\bigoplus_n m^n / m^{n+1}</math>
is isomorphic to a polynomial ring over ''k''. Broadly speaking, regular local rings are somewhat similar to polynomial rings.{{sfn|Matsumura|1989|p=143|loc=§7, Remarks|ps=}} Regular local rings are UFD's.{{sfn|Matsumura|1989|loc=§19, Theorem 48|ps=}}

[[Discrete valuation ring]]s are equipped with a function which assign an integer to any element ''r''. This number, called the valuation of ''r'' can be informally thought of as a zero or pole order of ''r''. Discrete valuation rings are precisely the one-dimensional regular local rings. For example, the ring of germs of holomorphic functions on a [[Riemann surface]] is a discrete valuation ring.

=== 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=}}

=== Cohen–Macaulay rings ===
The [[depth (ring theory)|depth]] of a local ring ''R'' is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence ''a''<sub>1</sub>, ..., 
''a''<sub>''n''</sub> &isin; ''m'' such that all ''a''<sub>''i''</sub> are non-zero divisors in
{{block indent|1= ''R'' / (''a''<sub>1</sub>, ..., ''a''<sub>''i''&minus;1</sub>). }}
For any local Noetherian ring, the inequality
{{block indent|1= depth(''R'') &le; dim(''R'') }}
holds. A local ring in which equality takes place is called a [[Cohen–Macaulay ring]]. Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being [[universally catenary ring]]s, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.{{sfn|Eisenbud|1995|loc=Corollary 18.10, Proposition 18.13|ps=}}