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.

Title local ring
Canonical name LocalRing
Date of creation 2013-03-22 12:37:44
Last modified on 2013-03-22 12:37:44
Owner djao (24)
Last modified by djao (24)
Numerical id 13
Author djao (24)
Entry type Definition
Classification msc 16L99
Classification msc 13H99
Classification msc 16L30
Related topic DiscreteValuationRing
Related topic LocallyRingedSpace
Related topic SemiLocalRing
Defines local ring homomorphism