In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

Definition and first consequences

A ring R is local if it has one (and therefore all) of the following equivalent properties: In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

• 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 a 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 obtain, 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 forth 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 I₁, I₂ where two ideals are called coprime if their sum of ideals I=I₁+I₂ is equal to R. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. It is denoted by J(R) and can be defined in the following equivalent ways: the... In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... A principal is: The head of an institution. ...

In the case of commutative rings 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. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Some authors require that a local ring be (left and right) Noetherian, and the non-Noetherian rings are then called "quasi-local". In this encyclopedia this requirement is not imposed. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...

Examples

Commutative

All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ...

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 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. In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ...

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 schemes, the generalizations of varieties, are defined as special locally ringed spaces. In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

A more arithmetical example is the following: the ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. 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. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, any integer (whole number) is either even or odd. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ...

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. In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

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 variety of dualities in mathematics are listed at duality (mathematics). ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... The word modulo is the Latin ablative of modulus. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...

Local rings play a major role in valuation theory. Given a field K, we may look for local rings in it on the assumption that it is a function field. 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 a function field of an algebraic variety V, then for each point P of V we can 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 Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...

F(P) = G(P) = 0,

the function

F/G

is an indeterminate form at P. Considering a simple example such as In mathematics, a number of the expressions that may be encountered in calculus and occasionally elsewhere are considered to be indeterminate forms, and must be treated as symbolic only, until more careful discussion has taken place. ...

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 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. In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. ... In abstract algebra, the length of a module is a measure of the modules size. It is defined as the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. ...

If k is a field of characteristic p > 0 and G is a finite p-group, then the group algebra kG is local. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ...

Some facts and definitions

Commutative

We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ring 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. 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 maps R × R → R, where R × R carries the product topology. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... 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 maps R × R → R, where R × R carries the product topology. ...

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. In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

As for any topological ring, one can ask whether (R,m) is complete; if it is not, one considers its completion, again a local ring. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

If (R,m) is a commutative Noetherian local ring, then In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...

(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

General

The Jacobson radical m of a local ring R (which is equal to the unique left maximal ideal and also to the unique right maximal ideal) consists precisely of the non-units of the ring; furthermore it is the unique two-sided maximal ideal of R. (In the non-commutative case, having a unique two-sided maximal is however not equivalent to being local.) In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. It is denoted by J(R) and can be defined in the following equivalent ways: the...

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 I ≠ R is any two-sided ideal in R, then the factor ring R/I is again local, with maximal ideal m/I. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ...

A deep theorem by Irving Kaplansky says that any projective module over a local ring is free. 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 (that is, a module with basis vectors). ... In mathematics, a free module is a module having a free basis. ...

See also

• semi-local ring