# Semi-local ring

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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*. Template:HarvTemplate:Harv

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 is a semi-local ring. In particular, if is a prime power, then is a local ring.
- A finite direct sum of fields 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*m*_{1}, ..., m_{n}

- .
- (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*, where the_{i})*p*are finitely many prime ideals._{i}

## Textbooks

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