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单词 LocalRing
释义

local ring


Commutative case

A commutative ring with multiplicative identityPlanetmathPlanetmath is called local if it has exactly one maximal idealMathworldPlanetmath.This is the case if and only if 10 and the sum of any two non-units (http://planetmath.org/unit) in the ring is again a non-unit; the unique maximal ideal consists precisely of the non-units.

The name comes from the fact that these rings are important in the study of the local behavior of varietiesMathworldPlanetmathPlanetmath (http://planetmath.org/variety) and manifolds: the ring of function germs at a point is always local. (The reason is simple: a germ f is invertiblePlanetmathPlanetmath in the ring of germs at x if and only if f(x)0, which implies that the sum of two non-invertible elements is again non-invertible.)This is also why schemes, the generalizationsPlanetmathPlanetmath of varieties, are defined as certain locally ringed spaces. Other examples of local ringsMathworldPlanetmath include:

  • All fields are local. The unique maximal ideal is (0).

  • Rings of formal power series over a field are local, even in several variables. The unique maximal ideal consists of those without .

  • if R is a commutative ring with multiplicative identity, and 𝔭 is a prime idealMathworldPlanetmathPlanetmath in R, then the localizationMathworldPlanetmath of R at 𝔭, written as R𝔭, is always local. The unique maximal ideal in this ring is 𝔭R𝔭.

  • All discrete valuation rings are local.

A local ring R with maximal ideal 𝔪 is also written as (R,𝔪).

Every local ring (R,𝔪) is a topological ring in a natural way, taking the powers of 𝔪 as a neighborhood base of 0.

Given two local rings (R,𝔪) and (S,𝔫), a local ring homomorphism from R to S is a ring homomorphismMathworldPlanetmath f:RS (respecting the multiplicative identities) with f(𝔪)𝔫. These are precisely the ring homomorphisms that are continuousPlanetmathPlanetmath with respect to the given topologiesMathworldPlanetmath on R and S.

The residue fieldMathworldPlanetmath of the local ring (R,𝔪) is the field R/𝔪.

General case

One also considers non-commutative local rings. A ring (http://planetmath.org/ring) with multiplicative identity is called local if it has a unique maximal left idealMathworldPlanetmathPlanetmath. In that case, the ring also has a unique maximal right ideal, and the two coincide with the ring’s Jacobson radicalMathworldPlanetmath, which in this case consists precisely of the non-units in the ring.

A ring R is local if and only if the following condition holds: we have 10, and whenever xR is not invertible, then 1-x is invertible.

All skew fields are local rings. More interesting examples are given by endomorphism ringsMathworldPlanetmath: a finite-length module over some ring is indecomposable if and only if its endomorphism ring is local, a consequence of Fitting’s lemma.

Titlelocal ring
Canonical nameLocalRing
Date of creation2013-03-22 12:37:44
Last modified on2013-03-22 12:37:44
Ownerdjao (24)
Last modified bydjao (24)
Numerical id13
Authordjao (24)
Entry typeDefinition
Classificationmsc 16L99
Classificationmsc 13H99
Classificationmsc 16L30
Related topicDiscreteValuationRing
Related topicLocallyRingedSpace
Related topicSemiLocalRing
Defineslocal ring homomorphism
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更新时间:2025/5/4 16:34:50