In mathematics, more particularly in abstract algebra, 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 manifolds, or of algebraic number fields examined at a particular place, or prime. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, an absolute value is a function which measures the "size" of elements in a field or Integral domain. Local algebra is the branch of commutative algebra that studies local rings and their modules. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. ^[1] The English term local ring is due to Zariski. Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July ^[2]

Contents 1 Definition and first consequences 2 Examples 2.1 Commutative 2.2 Non-commutative 3 Some facts and definitions 3.1 Commutative 3.2 General 4 Notes 5 See also

Definition and first consequences

A ring R is a local ring if it has any one of the following equivalent properties:

• R has a unique maximal left ideal. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.
• R has a unique maximal right ideal.
• 1 ≠ 0 and the sum of any two non-units in R is a non-unit. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i
• 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. In Ring theory, a branch of Abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements 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 I₁, I₂ where two ideals are called coprime if R = I₁ + I₂. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than 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

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 has the commutative property

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

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 In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, an 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 In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a 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". In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable 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. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety 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 scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on

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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the parity of an object states whether it is even or odd 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 Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property 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 In Mathematics, the constant term of a Polynomial is the term of degree 0

The algebra of dual numbers over any field is local. A variety of dualities in mathematics are listed at Duality (mathematics. 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 Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, a geometric series is a series with a constant ratio between successive terms. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure In these cases elements are either nilpotent or invertible. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Local rings play a major role in valuation theory. Valuation in mathematics may refer to Valuation (algebra Valuation (logic Valuation (measure theory Given a field K, which may or may not be a function field, 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. In Abstract algebra, a valuation ring is an Integral domain D such that for every element x of its Field of fractions F 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. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety 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. In Calculus and other branches of Mathematical analysis, an indeterminate form is an Algebraic expression obtained in the context of Limits 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 rings in the study of direct sum decompositions of modules over some other rings. In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars 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, the length of a module is a measure of the module's "size"

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 In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach

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. In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the This is the m-adic topology on R. In Commutative algebra, the term completion refers to several related Functors on Topological rings and modules

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. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S.

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

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 Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological 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. In Mathematics, the residue field is a basic construction in Commutative algebra.

General

The Jacobson radical 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. In Ring theory, a branch of Abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements (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. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible 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. Irving Kaplansky ( March 22, 1917 &ndash June 25, 2006) was a Canadian Mathematician. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Mathematics, Nakayama's lemma is an important technical lemma in Commutative algebra and Algebraic geometry.

Notes

1. ^ Krull, Wolfgang (1938). "Dimensionstheorie in Stellenringen" (in German). J. Reine Angew. Math. 179: 204.  
2. ^ Zariski, Oscar (May 1943). Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July "Foundations of a General Theory of Birational Correspondences". Trans. Amer. Math. Soc. 53 (3): 497.  

See also

• Semi-local ring
• Valuation ring

In Mathematics, a semi-local ring is a ring with a finite number of Maximal ideals It must be said that some refer to such a ring in general as a Quasi-local In Abstract algebra, a valuation ring is an Integral domain D such that for every element x of its Field of fractions F

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