A commutative ring with multiplicative identity is called local if it has exactly one maximal ideal. This is the case if and only if and the sum of any two non-units (http://planetmath.org/unit) in the ring is again a non-unit; the unique maximal ideal consists precisely of the non-units.
The name comes from the fact that these rings are important in the study of the local behavior of varieties (http://planetmath.org/variety) and manifolds: the ring of function germs at a point is always local. (The reason is simple: a germ is invertible in the ring of germs at if and only if , which implies that the sum of two non-invertible elements is again non-invertible.) This is also why schemes, the generalizations of varieties, are defined as certain locally ringed spaces. Other examples of local rings include:
A local ring with maximal ideal is also written as .
Given two local rings and , a local ring homomorphism from to is a ring homomorphism (respecting the multiplicative identities) with . These are precisely the ring homomorphisms that are continuous with respect to the given topologies on and .
The residue field of the local ring is the field .
One also considers non-commutative local rings. A ring (http://planetmath.org/ring) with multiplicative identity is called local if it has a unique maximal left ideal. In that case, the ring also has a unique maximal right ideal, and the two coincide with the ring’s Jacobson radical, which in this case consists precisely of the non-units in the ring.
A ring is local if and only if the following condition holds: we have , and whenever is not invertible, then is invertible.
All skew fields are local rings. More interesting examples are given by endomorphism rings: a finite-length module over some ring is indecomposable if and only if its endomorphism ring is local, a consequence of Fitting’s lemma.
|Date of creation||2013-03-22 12:37:44|
|Last modified on||2013-03-22 12:37:44|
|Last modified by||djao (24)|
|Defines||local ring homomorphism|