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 (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 varieties (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 invertible in the ring of germs at $x$ 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:

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All fields are local. The unique maximal ideal is $(0)$ .
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Rings of formal power series over a field are local, even in several variables. The unique maximal ideal consists of those without .
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if $R$ is a commutative ring with multiplicative identity, and $\mathfrak{p}$ is a prime ideal in $R$ , then the localization of $R$ at $\mathfrak{p}$ , written as $R_{\mathfrak{p}}$ , is always local. The unique maximal ideal in this ring is $\mathfrak{p}R_{\mathfrak{p}}$ .
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All discrete valuation rings are local.

A local ring $R$ 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 $R$ to $S$ 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 $R$ and $S$ .

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 (http://planetmath.org/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 coincide with the ring’s Jacobson radical, 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 $1\not=0$ , and whenever $x\in R$ is not invertible, then $1-x$ 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.

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