Regular local ring
In the mathematical subfield of commutative algebra is meant by a regular local ring has a Noetherian local ring whose maximal ideal of elements can be generated when the dimension of the ring respectively. Regular local rings describe the behavior of algebraic- geometric objects at points where there are no singularities as spikes or crossovers. A (not necessarily local ) ring is called regular if all its localizations are regular local rings.
This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.
Definition
It should be a -dimensional Noetherian local ring with maximal ideal and residue field. Then is called regular if one of the following equivalent conditions is satisfied:
- Can be formed of elements.
An arbitrary Noetherian ring is called regular if all its local rings are regular.
Properties
- Regular local rings are unique factorization.
- Criterion of Serre: A Noetherian local ring is regular if its global dimension is finite.
- From the criterion of Serre follows: localizations of regular local rings are again regular.
Examples
- Artinian local rings are regular if and only if they are bodies.
- Commutative Algebra
- Ring ( algebra)