Commutative case

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 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 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:

• All fields are local. The unique maximal ideal is .
• Rings of formal power series over a field are local, even in several variables. The unique maximal ideal consists of those power series without constant term.
• if is a commutative ring with multiplicative identity, and is a prime ideal in , then the localization of at , written as , is always local. The unique maximal ideal in this ring is .
• All discrete valuation rings are local.

A local ring with maximal ideal is also written as .

Every local ring is a topological ring in a natural way, taking the powers of as a neighborhood base of 0.

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 .

General case

One also considers non-commutative local rings. A 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 ideals 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.