Local ring

A local ring is in the mathematical field of ring theory, a ring in which there is exactly one maximal left or right ideal. Local rings play in algebraic geometry an important role to describe the "local " behavior of functions on algebraic varieties and manifolds.

The concept of the local ring was introduced in 1938 by Wolfgang Krull.

Definition

A ring R with 1 is called local if it satisfies one of the following equivalent conditions:

  • R has exactly one maximal left ideal.
  • R has exactly one maximal right ideal.
  • 1 ≠ 0 and each sum of two non- units is a unit.
  • 1 ≠ 0 and for any non- unit x 1- x is a unit.
  • If a finite sum of ring members is a moiety, then at least one summand is a unit ( in particular, the sum is not a blank unit, so it follows 1 ≠ 0).

Some authors require that a local ring must be additionally Noetherian, and call a non- Noetherian ring with exactly one maximal left ideal quasi locally. Here we omit this additional requirement and may be explicitly speak of Noetherian local rings.

Properties

Is local, then

Commutative case

If the ring R is commutative with 1, then the following additional terms equivalent to the locality:

  • R has exactly one maximal (two-sided ) ideal.
  • The complement of the unit group is an ideal.

For the equivalence of the two latter conditions, a proof is given here:

  • Possession of the commutative ring R with 1 exactly one maximal ideal I, and let x be a ring element which is not in I. Suppose that x would not be invertible. Then the principal ideal of x generated a proper ideal. As a proper ideal xR is a subset of the (only ) maximal ideal I. Thus x would be an element of I, contradicting the choice of x. So x is invertible, and so is each element of the complement of I is invertible. Since no element of I is invertible, I is exactly the complement of the unit group.
  • The complement of the unit is now an ideal group I. Since each ideal, which is above I, containing a unit, and thus is already a source of ring I is a maximum ideal.

Examples

  • Every body and every division ring is a local ring, since { 0} is the only maximal ideal in it.
  • The ring of integers is not local. For example, -2 and 3 no units, but rather their sum 1
  • The maximal ideals of the residue class ring are the ideals generated by the residue classes of prime divisors of. The ring is so if and only locally if n is a prime power.
  • The set of all rational numbers, which have are an odd number in the denominator in reduced fraction representation, forms a subring of the rational numbers, which is a local ring. Its maximal ideal consists of all fractions, the numerator of which is straight. This ring to write as: and calls him the " localizing at 2 ". It is produced by a process which is called localization of a ring.
  • The ring of formal power series with coefficients in a field is a local ring. Its maximal ideal consists of the power series, starting with the linear term. The constant term always vanishes.
  • The factor ring of the polynomial ring over a field K modulo the ideal generated locally. Its maximal ideal consists of the residue classes of polynomials without constant term. In this ring, each element is either invertible or nilpotent. A special case of this form, the dual numbers, the elements of the factor ring. This algebra is a two-dimensional vector space over.

Localization of rings

Be an arbitrary commutative ring with 1 and a closed under multiplication subset is called with, then

Localization in.

If the complement of a prime ideal, then is a local ring and is quoted at.

Unit (ring theory) Empty sum Jacobson radical Indecomposable module Quotient ring Michiel Hazewinkel Encyclopedia of Mathematics
527687
de