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Concept
Local ring
Summary
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
Definition and first consequences
A ring R is a local ring if it has any one of the following equivalent properties:
- R has a unique maximal left ideal.
- R has a unique maximal right ideal.
- 1 ≠ 0 and the sum of any two non-units in R is a non-unit.
- 1 ≠ 0 and if x is any element of R, then x or
Official source
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