Editing Regular local ring (section)

==Examples==
# Every [[field (mathematics)|field]] is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
# Any [[discrete valuation ring]] is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. For example, if ''k'' is a field and ''X'' is an indeterminate, then the ring of [[formal power series]] ''k''{{brackets|''X''}} is a regular local ring having (Krull) dimension 1.
# If ''p'' is an ordinary prime number, the ring of [[p-adic integer]]s is an example of a discrete valuation ring, and consequently a regular local ring. In contrast to the example above, this ring does not contain a field.
# More generally, if ''k'' is a field and ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub> are indeterminates, then the ring of formal power series ''k''{{brackets|''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub>}} is a regular local ring having (Krull) dimension ''d''.
# Still more generally, if ''A'' is a regular local ring, then the [[formal power series]] ring ''A''{{brackets|''x''}} is regular local.
# If '''Z''' is the ring of integers and ''X'' is an indeterminate, the ring '''Z'''[''X'']<sub>(2, ''X'')</sub> (i.e. the ring '''Z'''[''X''] [[Localization of a ring and a module|localized]] in the prime ideal (2, ''X'') ) is an example of a 2-dimensional regular local ring which does not contain a field.
# By the [[cohen structure theorem|structure theorem]] of [[Irvin Cohen]], a [[completion (ring theory)|complete]] regular local ring of Krull dimension ''d'' that contains a field ''k'' is a power series ring in ''d'' variables over an [[extension field]] of ''k''.