Semi-local ring

{{for|the older meaning of a Noetherian ring with a topology defined by an ideal in the Jacobson radical |Zariski ring}}

In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. {{harv|Lam|2001|p=§20}}{{harv|Mikhalev|Pilz|2002|p=C.7}}

The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".

Some literature refers to a commutative semi-local ring in general as a

quasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals.

A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.

Examples

Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
The quotient $\mathbb{Z}/m\mathbb{Z}$ is a semi-local ring. In particular, if $m$ is a prime power, then $\mathbb{Z}/m\mathbb{Z}$ is a local ring.
A finite direct sum of fields $\bigoplus_{i=1}^n{F_i}$ is a semi-local ring.
In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m₁, ..., m_n

: $R/\bigcap_{i=1}^n m_i\cong\bigoplus_{i=1}^n R/m_i\,$ .

:(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩_i m_i=J(R), and we see that R/J(R) is indeed a semisimple ring.

The classical ring of quotients for any commutative Noetherian ring is a semilocal ring.
The endomorphism ring of an Artinian module is a semilocal ring.
Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p_i), where the p_i are finitely many prime ideals.

Textbooks

{{citation

|last=Lam |first=T.Y.

|title=A first course in noncommutative rings

|series=Graduate Texts in Mathematics

|volume=131

|edition=2

|publisher=Springer-Verlag

|place=New York

|date=2001

|chapter=7

|pages=xx+385

|isbn=0-387-95183-0

|mr=1838439

}}

{{citation

|title=The concise handbook of algebra

|editor1-last=Mikhalev |editor1-first=Alexander V.

|editor2-last=Pilz |editor2-first=Günter F.

|publisher=Kluwer Academic Publishers

|place=Dordrecht

|date=2002

|pages=xvi+618

|isbn=0-7923-7072-4

|mr=1966155

}}

Category:Ring theory

Category:Localization (mathematics)