Local ring

In the mathematical field of ring theory, a local ring is a ring in which there is exactly one maximum left or right ideal . Local rings play an important role in algebraic geometry 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 under the name "Stellenringe".

definition

A ring with is called local if it fulfills one of the following equivalent conditions: ${\displaystyle R}$ ${\displaystyle 1}$

• ${\displaystyle R}$ has exactly one maximum left ideal.
• ${\displaystyle R}$ has exactly one maximum legal ideal.
• ${\displaystyle 1\neq 0}$ and every sum of two non-units is a non-unit.
• ${\displaystyle 1\neq 0}$and for every non- unit there is a unit . ${\displaystyle x}$ ${\displaystyle 1-x}$
• If a finite sum of ring elements is a unit, then at least one summand is a unit (in particular the empty sum is not a unit, so it follows from this ). ${\displaystyle 1\neq 0}$

Some authors require that a local ring must also be Noetherian , and call a non-Noetherian ring with exactly one maximum left ideal quasilocal . Here we omit this additional requirement and, if necessary, speak explicitly of Noetherian local rings.

properties

Is local, then ${\displaystyle R}$

1. the maximum left ideal coincides with the maximum right ideal and with the Jacobson radical . ${\displaystyle J}$
2. is a skew field (referred to as the remainder class field ), ${\displaystyle R/J}$
3. R has only the trivial idempotents and . As a - module cannot be dismantled . ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle R}$ ${\displaystyle R}$
4. is also semi-perfect . ${\displaystyle R}$

Commutative case

If the ring is commutative with 1, then the following conditions are also equivalent to the locality: ${\displaystyle R}$

• ${\displaystyle R}$ has exactly one maximum (both-sided) ideal.
• The complement of the unit group is an ideal. ${\displaystyle R^{*}}$

Proof of the equivalence of the last two conditions is given here:

• Have the commutative ring with exactly one maximum ideal , and be a ring element that is not in . Assuming it would not be invertible. Then the main ideal generated by is a real ideal. As a real ideal is a subset of the (single) maximal ideal . Thus an element of would be contrary to the choice of . So is invertible, and thus every element of the complement of is invertible. Since no element of is invertible, it is exactly the complement of the unit group. ${\displaystyle 1}$ ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle xR}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle I}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle I}$
• Now let the complement of the unit group be an ideal . Since every ideal that lies above contains a unity and is therefore already the whole ring, there is a maximum ideal. ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle I}$

Examples

Local rings in algebra

• Every body and every oblique body is a local ring since the only maximal ideal is in it. ${\displaystyle \{0\}}$
• Rating rings are local rings.
• The ring of integers is not local. For example, and are not units, but their sum is . ${\displaystyle \mathbb {Z} }$ ${\displaystyle -2}$ ${\displaystyle 3}$ ${\displaystyle 1}$
• The maximal ideals of the residue class ring are those of the residue classes of prime divisors of generated ideals. The ring is local if and only if there is a prime power. ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle n}$ ${\displaystyle n}$
• The set of all rational numbers , which have an odd number in the denominator when the fraction is abbreviated , forms a subring of the rational numbers, which is a local ring. Its maximum ideal consists of all fractions whose numerator is even. This ring is written as: and calls it the "localization of at ". It arises from a process called localization of a ring .
        ${\displaystyle \mathbb {Z} _{(2)}=\left\{\left.{\frac {a}{b}}\ \right\vert \ a,b\in \mathbb {Z} ,2\nmid b\right\}}$
  ${\displaystyle \mathbb {Z} }$ ${\displaystyle 2}$ ${\displaystyle \mathbb {Z} }$
• The ring of formal power series with coefficients in a field is a local ring. His maximum ideal consists of the power series that begin with the linear term. The constant term always disappears.
• The factor ring of the polynomial ring over a field modulo the ideal generated by is local. Its maximum ideal consists of the residual classes of the polynomials without an absolute term. In this ring, each element is either invertible or nilpotent . The dual numbers , the elements of the factor ring, form a special case of this . This algebra is two-dimensional over as a vector space . ${\displaystyle K[X]/(X^{n})}$ ${\displaystyle K}$ ${\displaystyle X^{n}}$ ${\displaystyle K[X]/(X^{2})}$ ${\displaystyle K}$

The seeds of continuous functions

Let be a point in a manifold , e.g. B. . On the set of (any) in environments of defined continuous functions we define an equivalence relation in that two (possibly different) in environments defined functions should be equivalent if there is a around there that are defined on the both functions and match. The equivalence classes of this relation are called seeds . The addition and multiplication of seeds are well defined. The set of seeds of continuous functions in forms a local ring, the maximum ideal of which is formed by the seeds of the in vanishing continuous functions. ${\displaystyle x}$ ${\displaystyle M}$ ${\displaystyle M=\mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x}$

Local rings of an algebraic variety

Let be an algebraic variety and . The local ring is defined as the amount of the germs normal functions in . It is a local ring, the maximum ideal of which forms the seeds of the regular functions that vanish in. It is obtained as a localization of the coordinate ring at the corresponding maximum ideal : ${\displaystyle V}$ ${\displaystyle x\in V}$ ${\displaystyle {\mathcal {O}}_{x}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle k\left[V\right]}$ ${\displaystyle x}$ ${\displaystyle {\mathfrak {m}}_{x}}$

${\displaystyle {\mathcal {O}}_{x}=k\left[V\right]_{{\mathfrak {m}}_{x}}}$.

The local dimension of in is defined as the Krull dimension of the local ring : ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {O}}_{x}}$

${\displaystyle \dim _{x}V:=\dim {\mathcal {O}}_{x}}$.

Localization of rings

→ Main article : Localization (algebra)

Let be any commutative ring with and a subset closed under multiplication with , then is called ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle S}$ ${\displaystyle 1\in S,\ 0\not \in S}$

${\displaystyle S^{-1}R:=\left\{{\frac {r}{s}}:r\in R,s\in S\right\}}$

the localization of in . ${\displaystyle R}$ ${\displaystyle S}$

If the complement is a prime ideal , then it is a local ring and is noted with . ${\displaystyle S:=R-p}$ ${\displaystyle p\subset R}$ ${\displaystyle S^{-1}R}$ ${\displaystyle R_{p}}$

literature

• VI Danilov: Local ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).