Editing Commutative ring (section)

=== 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.