Local ring

In abstract algebra, the local rings are certain rings comparatively simple and that serve to describe the local behavior of functions defined on algebraic varieties or differentiable varieties.

Definition and first consequences

R is a local ring if it satisfies the following equivalent properties:

• R has a unique ideal on the maximal left
• R has a unique ideal for the maximal right
• 1 and the sum of any pair of elements in R that are not units is not a unit either
• 1 and yes x is any element of R, then x or 1-x It's a unit.
• If a finite sum is a unit, then it will also be one of its sums.

If these properties are given, then the only maximal left ideal coincides with the only maximal right ideal and also with the Jacobson Radical of the ring.

In the case of commutative rings it is not necessary to distinguish between ideals on one side or the other, so a commutative ring is local if and only if it has a unique maximal ideal.

Some authors define a local ring as requiring it to be noetherian, with non-noetherians being called quasi-local rings. Wikipedia will not use this latter definition of local ring.

Examples

Commutatives

All fields (and "skew" fields, ie division rings) are local rings, since {0} is the only maximal ideal in such rings.

To motivate the name of "locales" for such rings, let us consider real continuous functions defined on some open interval around 0 on the real line. we will be interested only on the local behavior of such functions near 0 and we will identify two functions if they coincide over some (possibly very small) open interval around 0. This identification defines an Equivalence Relation, and the Equivalence Classes are the "seeds of real-valued functions at 0". Such seeds can be added and multiplied and have the ring structure.

To see that this seed ring is local, we need to identify its invertible elements. A seed f is invertible if f(0) ≠ 0. The reason: if f(0) ≠ 0, then there exists an open interval around 0 where f is non-zero, and we can form the function g(x) = 1 /f(x) over that interval. The function g then gives us another seed, and the product of fg equals 1.

With this characterization, it is clear that the sum of any two non-invertible seeds is again non-invertible, and we have a local commutative ring. The maximal ideal of such a ring is composed precisely of those seeds f such that f(0) = 0.

The same argument works for the ring of seeds of real-valued functions on any topological space at a given point, or the ring of seeds of differentiable functions on any differentiable Manifold at a given point, or the ring of seeds of rational functions on any algebraic Manifold at a given point. All these rings are therefore local. These examples explain why schemata, the generalization of manifolds, are defined as special types of locally ringed Space.

A more arithmetic example is the following: the ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with an even numerator and an odd denominator. More generally, given any commutative ring R and any Prime Ideal P of R, the location of R in P is local; the maximal ideal is the ideal generated by P at this location.

Every formal Power Series ring over a field (even over several variables) is local; the maximal ideal consists of those power series with no constant term.

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 and its maximal ideal consists of the classes of polynomials with nonzero constant term.

Local rings play a fundamental role in Valuation theory. Given a field K, we can search for local rings in it under the assumption that it is a field of functions. By definition a valuation ring of K is a subring R, such that for every nonzero element x of K, either x is in R or x^{-1 is} . Any such subring will be a local ring. If K were really a field of functions of an algebraic Manifold V, then for each point P of V we can try to define a ring of valuation R of functions defined on P. In cases where V has dimension 2 or greater there is a difficulty: if F and G are rational functions on V with F(P) = G(P) = 0, the function F/G is an indeterminate Form in P. Considering a simple example such as Y/X, approximating along a line Y=tX, we see that the value in P is a concept that it lacks a simplistic definition, and this is obtained through the use of valuations.

Noncommutativity

Local non-commutative rings arise naturally as rings of endomorphisms in the study of decompositions into direct sums of modules over other rings. Specifically, if the ring of endomorphisms of the module M is local, then M is non-decomposable; and vice versa, if the module M has finite length and is non-decomposable, then its ring of endomorphisms is local.

If k is a field of feature p > 0 and G is a finite p-group, then the algebra of groups kG it's local.

Some aspects and definitions

Commutativity

We will write (R, m) to denote a local commutative ring R with maximal ideal m. Such a ring forms a topological Ring in a natural way if we take the powers of m as a Neighbor Basis of 0. This is called 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. Which are precisely the ring homomorphisms that are continuous with respect to the given topologies on R and S.

As for any topological ring, we can ask whether (R, m) is complete; if it is not, its complexion can again be considered to be a local ring.

If (R, m) is a commutative Noetherian local ring, then

i=1∞ ∞ mi={0!{displaystyle bigcap _{i=1}^{infty }m^{i}={0}}

(Krull's intersection theorem), and it follows R together with the m-adic topology is a Hausdorff Space.

General information

The Jacobson Radical m of a local ring R (which is equal to the only maximal ideal on the left and also to the only maximal ideal on the right) is formed precisely of the elements of the ring that are not units; furthermore, it is the only maximal ideal on both sides of R. (In the non-commutative case, having a single maximal ideal on both sides is not, however, to be local).

Let x be an element of the local ring R, the following statements are equivalent:

• x has an inverse on the left
• x has an inverse on the right
• x It's invertible.
• x He's not in. m.

If (R, m) is local, then Ring factor R/m is a field "skew". If I ≠ R is any two-sided ideal in R, then the ring factor R/I is again local, with maximal ideal m/I.

A profound Kaplansky theorem says that any projective Modulus on a local ring is free.