What is "local ring"

n. 1 (context algebra English) A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique maximal right ideal. 2 (context networking English) The non-routing segment of a token ring network.

In abstract algebra, more specifically 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.

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 in 1938 under the name Stellenringe. The English term local ring is due to Zariski.