Local ring
Encyclopedia
In abstract algebra
, more particularly in ring theory
, local rings are certain rings
that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties
or manifold
s, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra
that studies local rings and their modules
.
The concept of local rings was introduced by Wolfgang Krull
in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
R is a local ring if it has any one of the following equivalent properties:
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical
. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime
proper (principal
) (left) ideals where two ideals I1, I2 are called coprime if R = I1 + I2.
In the case of commutative ring
s, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Some authors require that a local ring be (left and right) Noetherian
, and the non-Noetherian rings are then called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an integral domain is called a local domain.
s (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
s, which are local principal ideal domain
s that are not fields.
Every ring of formal power series
over a field (even in several variables) is local; the maximal ideal consists of those power series without constant term
.
Similarly, the algebra of dual numbers over any field is local. More generally, if F is a field 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 zero constant term, since one can use a geometric series to invert all other polynomials modulo
Xn. In these cases elements are either nilpotent
or invertible.
A more arithmetical example is the following: the ring of rational number
s with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator: this 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.
s defined on some open interval
around 0 of the real line
. We are only interested in the local behavior of these functions near 0 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 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 identify its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1.
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 scheme
s, the generalizations of varieties, are defined as special locally ringed spaces.
, we may look for local rings in it. By definition, a valuation ring
of 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. 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
the function
is an indeterminate form
at P. Considering a simple example, such as
approached along a line
one sees that the value at P is a concept without a simplistic definition. It is replaced by using valuations.
s 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
p > 0 and G is a finite p-group
, then the group algebra
kG is local.
in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R.
If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism
f : R → S with the property f(m) ⊆ n. These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S.
A ring morphism f : R → S is a local ring homomorphism if and only if
; that is, the preimage of the maximal ideal is maximal.
As for any topological ring, one can ask whether (R, m) is complete (as a topological space); if it is not, one considers its completion
, again a local ring.
If (R, m) is a commutative Noetherian
local ring, then
(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space
.
In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field
of the local ring or residue field of the point P.
m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.
For an element x of the local ring R, the following are equivalent:
If (R, m) is local, then the factor ring R/m is a skew field. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J.
A deep theorem by Irving Kaplansky
says that any projective module
over a local ring is free
, though the case where the module is finitely-generated is a simple corollary to Nakayama's lemma. This has an interesting consequence in terms of Morita equivalence
. Namely, if P is a finitely generated projective R module, then P is isomorphic to the free module Rn, and hence the ring of endomorphisms
is isomorphic to the full ring of matrices 
. Since every ring Morita equivalent to the local ring R is of the form 
for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, more particularly in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, local rings are certain rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
or manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
that studies local rings and their modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
.
The concept of local rings was introduced by Wolfgang Krull
Wolfgang Krull
Wolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in Baden-Baden, Imperial Germany and died in Bonn, West Germany.- See also :* Krull dimension* Krull topology...
in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
Definition and first consequences
A ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R is a local ring if it has any one of the following equivalent properties:
- R has a unique maximal left ideal.
- R has a unique maximal right ideal.
- 1 ≠ 0 and the sum of any two non-units in R is a non-unit.
- 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.
- If a finite sum is a unit, then so are some of its terms (in particular the empty sum is not a unit, hence 1 ≠ 0).
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
proper (principal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
) (left) ideals where two ideals I1, I2 are called coprime if R = I1 + I2.
In the case of commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Some authors require that a local ring be (left and right) Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, and the non-Noetherian rings are then called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an integral domain is called a local domain.
Fields
All fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
s (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
Discrete valuation rings
An important class of local rings are discrete valuation ringDiscrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
s, which are local principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
s that are not fields.
Polynomial
Every ring of formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
over a field (even in several variables) is local; the maximal ideal consists of those power series without constant term
Constant term
In mathematics, a constant term is a term in an algebraic expression has a value that is constant or cannot change, because it does not contain any modifiable variables. For example, in the quadratic polynomialx^2 + 2x + 3,\ the 3 is a constant term....
.
Similarly, the algebra of dual numbers over any field is local. More generally, if F is a field and n is a positive integer, then the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
F[X]/(Xn) is local with maximal ideal consisting of the classes of polynomials with zero constant term, since one can use a geometric series to invert all other polynomials modulo
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
Xn. In these cases elements are either nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
or invertible.
Arithmetic
A more arithmetical example is the following: the ring of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator: this is the integers localized
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
at 2.
More generally, given any commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R and any prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization.
Ring of germs
To motivate the name "local" for these rings, we consider real-valued continuous functionContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s defined on some open interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
around 0 of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
. We are only interested in the local behavior of these functions near 0 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
, and the equivalence classes are the "germs
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
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 identify its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1.
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
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
at a given point, or the ring of germs of differentiable functions on any differentiable manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
at a given point, or the ring of germs of rational functions on any algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
at a given point. All these rings are therefore local. These examples help to explain why scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s, the generalizations of varieties, are defined as special locally ringed spaces.
Valuation theory
Local rings play a major role in valuation theory. Given a field K, which may or may not be a function fieldFunction field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
, we may look for local rings in it. By definition, a valuation ring
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
of 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. If K were indeed the function field of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
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
Indeterminate form
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...
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 simplistic definition. It is replaced by using valuations.
Non-commutative
Non-commutative local rings arise naturally as endomorphism ringEndomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
s in the study of direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
decompositions of modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...
; conversely, if the module M has finite length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
and is indecomposable, then its endomorphism ring is local.
If k is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p > 0 and G is a finite p-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
, then the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
kG is local.
Commutative
We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ringTopological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R.
If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
f : R → S with the property f(m) ⊆ n. These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S.
A ring morphism f : R → S is a local ring homomorphism if and only if
; that is, the preimage of the maximal ideal is maximal.As for any topological ring, one can ask whether (R, m) is complete (as a topological space); if it is not, one considers its completion
Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
, again a local ring.
If (R, m) is a commutative Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
local ring, then
(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
of the local ring or residue field of the point P.
General
The Jacobson radicalJacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.
For an element x of the local ring R, the following are equivalent:
- x has a left inverse
- x has a right inverse
- x is invertible
- x is not in m.
If (R, m) is local, then the factor ring R/m is a skew field. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J.
A deep theorem by Irving Kaplansky
Irving Kaplansky
Irving Kaplansky was a Canadian mathematician.-Biography:He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D...
says that any projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
over a local ring is free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
, though the case where the module is finitely-generated is a simple corollary to Nakayama's lemma. This has an interesting consequence in terms of Morita equivalence
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.- Motivation :...
. Namely, if P is a finitely generated projective R module, then P is isomorphic to the free module Rn, and hence the ring of endomorphisms
is isomorphic to the full ring of matrices . Since every ring Morita equivalent to the local ring R is of the form for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.