Local ring

In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. Local algebra is the branch of commutative algebra that studies local rings and their modules.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A ring R is a local ring if it has one (and therefore all) 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 a 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, 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.

Local ring: Definition and first consequences ►



 

Abstract algebra: :This article is about the branch of mathematics. For other uses of the term "algebra" see algebra (disambiguation)....

Rings: redirect Ring...

Varieties: REDIRECT Variety...




Local ring related Images and Photos (experimental)

Ring	Local Hero	Ring a Ring a Roses	Local Honey Journal
Local H: 12 Angry Months	Local H: 12 Angry Months	Power Ring	Sundial Ring
Ring Necked Pheasant	Local Honey Mini Notebook	Ring in the New	ICP - Ring Master