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 $1\ne 0$ and the sum of any two nonunits (http://planetmath.org/unit) in the ring is again a nonunit; the unique maximal ideal consists precisely of the nonunits.
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 $f(x)\ne 0$, which implies that the sum of two noninvertible elements is again noninvertible.) 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 $(0)$.

•
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 $\U0001d52d$ is a prime ideal^{} in $R$, then the localization^{} of $R$ at $\U0001d52d$, written as ${R}_{\U0001d52d}$, is always local. The unique maximal ideal in this ring is $\U0001d52d{R}_{\U0001d52d}$.

•
All discrete valuation rings are local.
A local ring $R$ with maximal ideal $\U0001d52a$ is also written as $(R,\U0001d52a)$.
Every local ring $(R,\U0001d52a)$ is a topological ring in a natural way, taking the powers of $\U0001d52a$ as a neighborhood base of 0.
Given two local rings $(R,\U0001d52a)$ and $(S,\U0001d52b)$, a local ring homomorphism from $R$ to $S$ is a ring homomorphism^{} $f:R\to S$ (respecting the multiplicative identities) with $f(\U0001d52a)\subseteq \U0001d52b$. 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 $(R,\U0001d52a)$ is the field $R/\U0001d52a$.
General case
One also considers noncommutative 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 nonunits in the ring.
A ring $R$ is local if and only if the following condition holds: we have $1\ne 0$, and whenever $x\in R$ is not invertible, then $1x$ is invertible.
All skew fields are local rings. More interesting examples are given by endomorphism rings^{}: a finitelength 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  20130322 12:37:44 
Last modified on  20130322 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 