Local ring

In mathematics, more specifically 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 algebraic 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 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 in 1938 under the name Stellenringe. The English term local ring is due to Zariski.

A ring R is a local ring if it has any one of the following equivalent properties:

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. 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 proper (principal) (left) ideals, where two ideals I₁, I₂ are called coprime if R = I₁ + I₂.

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, 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 is called a local domain.

To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open interval around ${\displaystyle 0}$ of the real line. We are only interested in the behavior of these functions near ${\displaystyle 0}$ (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around ${\displaystyle 0}$. This identification defines an equivalence relation, and the equivalence classes are what are called the "germs of real-valued continuous functions at ${\displaystyle 0}$". These germs can be added and multiplied and form a commutative ring.