local ring

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 $1\not=0$ 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 $f$ is invertible in the ring of germs at $x$ if and only if $f(x)\not=0$ , 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 $(0)$ .
• 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 $R$ is a commutative ring with multiplicative identity, and $\mathfrak{p}$ is a prime ideal in $R$ , then the localization of $R$ at $\mathfrak{p}$ , written as $R_{\mathfrak{p}}$ , is always local. The unique maximal ideal in this ring is $\mathfrak{p}R_{\mathfrak{p}}$ .
• All discrete valuation rings are local.

A local ring $R$ with maximal ideal $\mathfrak{m}$ is also written as $(R,\mathfrak{m})$ .

Every local ring $(R,\mathfrak{m})$ is a topological ring in a natural way, taking the powers of $\mathfrak{m}$ as a neighborhood base of 0.

Given two local rings $(R,\mathfrak{m})$ and $(S,\mathfrak{n})$ , a local ring homomorphism from $R$ to $S$ is a ring homomorphism $f:R\to S$ (respecting the multiplicative identities) with $f(\mathfrak{m})\subseteq\mathfrak{n}$ . These are precisely the ring homomorphisms that are continuous with respect to the given topologies on $R$ and $S$ .

The residue field of the local ring $(R,\mathfrak{m})$ is the field $R/\mathfrak{m}$ .

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 $R$ is local if and only if the following condition holds: we have $1\not=0$ , and whenever $x\in R$ is not invertible, then $1-x$ 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.