Local ring

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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.

The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.[1] The English term local ring is due to Zariski.[2]

Contents

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-units in R is a non-unit.
  • 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.
  • If a finite sum is a unit, then so are some of its terms (in particular the empty sum is not a unit, hence 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. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,[3] 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 I1, I2 where two ideals are called coprime if R = I1 + I2.

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