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1	Manifold
1	Division Ring
1	Point
1	Continuous
1	Field
1	Summation
1	Variable
1	Even Number
1	Relation Theory
1	Logical Implication
1	Implication
1	Commutative Ring
1	Ring
1	Ideal
1	Commutative
2	Scheme
2	Algebraic Topology
2	Prime Ideal
2	Discrete Valuation Ring
2	Ring Homomorphism
2	Formal Power Series
3	Fractional Ideal Of Commutative Ring
3	Maximal Ideal
4	Localization
6	Germ
6	Jacobson Radical
6	Topological Ring
7	Behavior
7	Variety Of Groups
8	Ring Of Endomorphisms
9	Local Base
11	Length
11	Indecomposable Group
13	Locally Ringed Space
20	Fittings Lemma

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Local Ring

Commutative case

A commutative ring with multiplicative identity is called local if it has exactly one maximal ideal. This is the case if and only if $1\not=0$ and the sum of any two non-units 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 varieties and manifolds: the ring of function germs at a point is always local. (The reason is simple: a germ is invertible in the ring of germs at if and only if $f(x)\not=0$ , which implies that the sum of two non-invertible elements is again non-invertible.) This is also why schemes, the generalizations of varieties, are defined as certain locally ringed spaces. Other examples of local rings include:

All fields are local. The unique maximal ideal is .
Rings of formal power series over a field are local, even in several variables. The unique maximal ideal consists of those power series without constant term.
if is a commutative ring with multiplicative identity, and $\mathfrak{p}$ is a prime ideal in , then the localization of at $\mathfrak{p}$ , written as $R_{\mathfrak{p}}$ , is always local. The unique maximal ideal in this ring is $\mathfrak{p}R_{\mathfrak{p}}$ .
All discrete valuation rings are local.

A local ring with maximal ideal $\mathfrak{m}$ is also written as $(R,\mathfrak{m})$ .

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

Given two local rings $(R,\mathfrak{m})$ and $(S,\mathfrak{n})$ , a local ring homomorphism from to is a ring homomorphism $f:R\to S$ (respecting the multiplicative identities) with $f(\mathfrak{m})\subseteq\mathfrak{n}$ . These are precisely the ring homomorphisms that are continuous with respect to the given topologies on and .

The residue field of the local ring $(R,\mathfrak{m})$ is the field $R/\mathfrak{m}$ .

General case

One also considers non-commutative local rings. A ring with multiplicative identity is called local if it has a unique maximal left ideal. In that case, the ring also has a unique maximal right ideal, and the two ideals coincide with the ring's Jacobson radical, which in this case consists precisely of the non-units in the ring.

A ring is local if and only if the following condition holds: we have $1\not=0$ , and whenever $x\in R$ is not invertible, then is invertible.

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