What is local ring?

Local Ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies local rings and their modules.

Read more about Local Ring.

Some articles on local ring:

Azumaya Algebra
... for the case where R is a commutative local ring ... The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki ... An Azumaya algebra over a commutative local ring R is an R-algebra A that is free and of finite rank r as an R-module, such that the tensor product (where Ao is the ...
Henselian Ring - Definitions
... In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings ... A local ring R with maximal ideal m is called Henselian if Hensel's lemma holds ... A local ring is Henselian if and only if every finite ring extension is a product of local rings ...
Local Ring - Some Facts and Definitions - General
... The Jacobson radical m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring ... a unique maximal two-sided ideal is not equivalent to being local ... For an element x of the local ring R, the following are equivalent x has a left inverse x has a right inverse x is invertible x is not in m ...
Regular Local Ring
... In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension ... In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1.. ... A point x on a algebraic variety X is nonsingular if and only if the local ring of germs at x is regular ...

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