• All
fields (and
skew fields) are local rings, since {0} is the only maximal ideal in these rings. • The ring \mathbb{Z}/p^n\mathbb{Z} is a local ring ( prime, ). The unique maximal ideal consists of all multiples of . • More generally, a nonzero ring in which every element is either a unit or
nilpotent is a local ring. • An important class of local rings are
discrete valuation rings, which are local
principal ideal domains that are not fields. • The ring \mathbb{C}
x, whose elements are infinite series \sum_{i=0}^\infty a_ix^i where multiplications are given by (\sum_{i=0}^\infty a_ix^i)(\sum_{i=0}^\infty b_ix^i)=\sum_{i=0}^\infty c_ix^i such that c_n=\sum_{i+j=n}a_ib_j, is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero. • More generally, every ring of
formal power series over a local ring is local; the maximal ideal consists of those power series with
constant term in the maximal ideal of the base ring. • Similarly, the
algebra of
dual numbers over any field is local. More generally, if
F is a local ring and
n is a positive integer, then the
quotient ring F[
X]/(
Xn) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of
F, since one can use a
geometric series to invert all other polynomials
modulo Xn. If
F is a field, then elements of
F[
X]/(
Xn) are either
nilpotent or
invertible. (The dual numbers over
F correspond to the case .) • Nonzero quotient rings of local rings are local. • The ring of
rational numbers with
odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers
localized at 2. • More generally, given any
commutative ring R and any
prime ideal P of
R, the
localization of
R at
P is local; the maximal ideal is the ideal generated by
P in this localization; that is, the maximal ideal consists of all elements
a/
s with
a ∈
P and
s ∈
R -
P.
Non-examples • The
ring of polynomials K[x] over a field K is not local, since x and 1 - x are non-units, but their sum is a unit. • The ring of integers \Z is not local since it has a maximal ideal (p) for every prime p. • \Z/(
pq)\Z, where
p and
q are distinct prime numbers. Both (
p) and (
q) are maximal ideals here.
Ring of germs To motivate the name "local" for these rings, we consider real-valued
continuous functions defined on some
open interval around 0 of the
real line. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an
equivalence relation, and the
equivalence classes are what are called the "
germs of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring. To see that this ring of germs is local, we need to characterize its invertible elements. A germ f is invertible if and only if f(0)\neq 0. The reason: if f(0)\neq 0, then by continuity there is an open interval around 0 where f is non-zero, and we can form the function g(x)=\frac 1{f(x)} on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. (Conversely, if f is invertible, then there is some g such that f(0)g(0)=1, hence f(0)\neq 0.) With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs f with f(0)=0. Exactly the same arguments work for the ring of germs of continuous real-valued functions on any
topological space at a given point, or the ring of germs of
differentiable functions on any
differentiable manifold at a given point, or the ring of germs of
rational functions on any
algebraic variety at a given point. All these rings are therefore local. These examples help to explain why
schemes, the generalizations of varieties, are defined as special
locally ringed spaces.
Valuation theory Local rings play a major role in valuation theory. By definition, a
valuation ring of a field
K is a subring
R such that for every non-zero element
x of
K, at least one of
x and
x−1 is in
R. Any such subring will be a local ring. For example, the ring of
rational numbers with
odd denominator (mentioned above) is a valuation ring in \mathbb{Q}. Given a field
K, which may or may not be a
function field, we may look for local rings in it. If
K were indeed the function field of an
algebraic variety V, then for each point
P of
V we could try to define a valuation ring
R of functions "defined at"
P. In cases where
V has dimension 2 or more there is a difficulty that is seen this way: if
F and
G are rational functions on
V with :
F(
P) =
G(
P) = 0, the function :
F/
G is an
indeterminate form at
P. Considering a simple example, such as :
Y/
X, approached along a line :
Y =
tX, one sees that the
value at P is a concept without a simple definition. It is replaced by using valuations.
Non-commutative Non-commutative local rings arise naturally as
endomorphism rings in the study of
direct sum decompositions of
modules over some other rings. Specifically, if the endomorphism ring of the module
M is local, then
M is
indecomposable; conversely, if the module
M has finite
length and is indecomposable, then its endomorphism ring is local. If
k is a
field of
characteristic and
G is a finite
p-group, then the
group algebra kG is local. == Some facts and definitions ==