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.
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] 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,[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, where two ideals I1, I2 are called coprime if R = I1 + I2.
[4] 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.
To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open interval around 0 of the real line.
We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0.
These germs can be added and multiplied and form a commutative ring.
To see that this ring of germs is local, we need to characterize its invertible elements.
The function g gives rise to a germ, and the product of fg is equal to 1.
The maximal ideal of this ring consists precisely of those germs f with f(0) = 0.
Exactly the same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point.
These examples help to explain why schemes, the generalizations of varieties, are defined as special locally ringed spaces.
Local rings play a major role in valuation theory.
If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with the function is an indeterminate form at P. Considering a simple example, such as approached along a line one sees that the value at P is a concept without a simple definition.
Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local.
If k is a field of characteristic p > 0 and G is a finite p-group, then the group algebra kG is local.
This is the m-adic topology on R. If (R, m) is a commutative Noetherian local ring, then (Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space.
Indeed, let R be the ring of germs of infinitely differentiable functions at 0 in the real line and m be the maximal ideal
Complete Noetherian local rings are classified by the Cohen structure theorem.
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; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.
A deep theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated is a simple corollary to Nakayama's lemma.
This has an interesting consequence in terms of Morita equivalence.