In
abstract algebra, more specifically
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, local rings are certain
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on
varieties
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
or
manifolds, or of
algebraic number fields
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a fi ...
examined at a particular
place, or prime. Local algebra is the branch of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
that studies commutative local rings and their
modules.
In practice, a commutative local ring often arises as the result of the
localization of a ring at a
prime ideal.
The concept of local rings was introduced by
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
in 1938 under the name ''Stellenringe''.
[
] The English term ''local ring'' is due to
Zariski
, birth_date =
, birth_place = Kobrin, Russian Empire
, death_date =
, death_place = Brookline, Massachusetts, United States
, nationality = American
, field = Mathematics
, work_institutions = ...
.
[
]
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.
* ''R'' has a unique maximal right ideal.
* 1 ≠ 0 and the sum of any two non-
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
s in ''R'' is a non-unit.
* 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or is a unit.
* If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 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 mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
. 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
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
proper (
principal) (left) ideals, where two ideals ''I''
1, ''I''
2 are called ''coprime'' if .
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.
Before about 1960 many authors required that a local ring be (left and right)
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
is called a local domain.
Examples
*All
fields (and
skew field
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not l ...
s) are local rings, since is the only maximal ideal in these rings.
*The ring
is a local ring ( prime, ). The unique maximal ideal consists of all multiples of .
*More generally, a nonzero ring in which every element is either a unit or nilpotent is a local ring.
*An important class of local rings are
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
s, which are local
principal ideal domains that are not fields.
*The ring
, whose elements are infinite series
where multiplications are given by
such that
, is local. Its unique maximal ideal consists of all elements which are not invertible. In other words, it consists of all elements with constant term zero.
*More generally, every ring of
formal power series over a local ring is local; the maximal ideal consists of those power series with
constant term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial
:x^2 + 2x + 3,\
the 3 is a constant term.
After like terms are com ...
in the maximal ideal of the base ring.
*Similarly, the algebra of
dual numbers
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
over any field is local. More generally, if ''F'' is a local ring and ''n'' is a positive integer, then the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''F''
'X''(''X''
''n'') is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of ''F'', since one can use a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
to invert all other polynomials
modulo ''X''
''n''. If ''F'' is a field, then elements of ''F''
'X''(''X''
''n'') are either
nilpotent or
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
. (The dual numbers over ''F'' correspond to the case .)
*Nonzero quotient rings of local rings are local.
*The ring of
rational numbers with
odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers
localized at 2.
*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; that is, the maximal ideal consists of all elements ''a/s'' with a ∈ ''P'' and s ∈ ''R'' - ''P''.
Non-examples
*The ring of polynomials