local ring

local ring

Commutative case

A

commutative ring

with

multiplicative identity

is called

local

if it has exactly one

maximal ideal

.
This is the case if and only if

1

≠

0

and the sum of any two non-units (

http://planetmath.org/unit

) in the ring is again a non-unit; the unique maximal ideal consists precisely of the non-units.

The name comes from the fact that these rings are important in the study of the local

behavior

of

varieties

(

http://planetmath.org/variety

) and manifolds: the ring of

function

germs at a point is always local. (The reason is simple: a germ

f

is

invertible

in the ring of germs at

x

if and only if

f

⁢

(

x

)

≠

0

, which implies that the sum of two non-invertible

elements

is again non-invertible.)
This is also why schemes, the

generalizations

of varieties, are defined as certain locally ringed spaces. Other examples of

local rings

include:

•

All fields are local. The unique maximal ideal is

(

0

)

.

•

Rings of

formal power series

over a field are local, even in several variables. The unique maximal ideal consists of those without .

•

if

R

is a commutative ring with multiplicative identity, and

𝔭

is a

prime ideal

in

R

, then the

localization

of

R

at

𝔭

, written as

R

𝔭

, is always local. The unique maximal ideal in this ring is

𝔭

⁢

R

𝔭

.

•

All

discrete valuation rings

are local.

A local ring

R

with maximal ideal

𝔪

is also written as

(

R

,

𝔪

)

.

Every local ring

(

R

,

𝔪

)

is a

topological ring

in a natural way, taking the powers of

𝔪

as a

neighborhood base

of 0.

Given two local rings

(

R

,

𝔪

)

and

(

S

,

𝔫

)

, a

local ring homomorphism

from

R

to

S

is a

ring homomorphism

f

:

R

→

S

(respecting the multiplicative identities) with

f

⁢

(

𝔪

)

⊆

𝔫

. These are precisely the ring homomorphisms that are

continuous

with respect to the given

topologies

on

R

and

S

.

The

residue field

of the local ring

(

R

,

𝔪

)

is the field

R

/

𝔪

.

General case

One also considers

non-commutative

local rings. A ring (

http://planetmath.org/ring

) with multiplicative identity is called

local

if it has a unique

maximal

left ideal

. In that case, the ring also has a unique maximal

right ideal

, and the two coincide with the ring’s

Jacobson radical

, which in this case consists precisely of the non-units in the ring.

A ring

R

is local if and only if the following condition holds: we have

1

≠

0

, and whenever

x

∈

R

is not invertible, then

1

-

x

is invertible.

All

skew fields

are local rings. More interesting examples are given by

endomorphism rings

: a

finite-length module

over some ring is

indecomposable

if and only if its endomorphism ring is local, a

consequence

of Fitting’s lemma.

Title

local ring

Canonical name

LocalRing

Date of creation

2013-03-22 12:37:44

Last modified on

2013-03-22 12:37:44

Owner

djao (24)

Last modified by

djao (24)

Numerical id

13

Author

djao (24)

Entry type

Definition

Classification

msc 16L99

Classification

msc 13H99

Classification

msc 16L30

Related topic

DiscreteValuationRing

Related topic

LocallyRingedSpace

Related topic

SemiLocalRing

Defines

local ring homomorphism

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