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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.
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 1 − x is a unit.
If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if R = I1 + I2.
In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Before about 1960 many authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an integral domain is called a local domain.
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