local ringLocalRing2002-05-02 22:40:582006-03-26 09:46:22Definitionlocal ring homomorphism% this is the default PlanetMath preamble. as your knowledge
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\subsection*{Commutative case}
A commutative ring with multiplicative identity is called {\em 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-\PMlinkname{units}{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 \PMlinkname{varieties}{variety} 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:
\begin{itemize}
\item All fields are local. The unique maximal ideal is $(0)$.
\item Rings of formal power series over a field are local, even in several variables. The unique maximal ideal consists of those \PMlinkescapetext{power series} without \PMlinkescapetext{constant term}.
\item 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 $\mathfrak{p}R_{\mathfrak{p}}$.
\item All discrete valuation rings are local.
\end{itemize}
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 \emph{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 {\em residue field} of the local ring $(R,\mathfrak{m})$ is the field $R/\mathfrak{m}$.
\subsection*{General case}
One also considers non-commutative local rings. A \PMlinkname{ring}{ring} with multiplicative identity is called \emph{local} if it has a unique maximal left ideal. In that case, the ring also has a unique maximal right ideal, and the two \PMlinkescapetext{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.