Semi-local ring

Semi-local ring

In mathematics, a semi-local ring is a ring with a finite number of maximal ideals. It must be said that some refer to such a ring in general as a
quasi-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 ideal.

Examples

* A finite direct sum of fields igoplus_{i=1}^n{F_i} is a semilocal ring.
* In the case of commutative rings with unit, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semilocal commutative ring with unit "R" and maximal ideals "m1, ..., mn":"R" / ∩i mi ≅ ⊕i "R / mi". (The map is the natural projection). The right hand side is a direct sum of fields.
* Semilocal rings occur for example in algebraic geometry when a (commutative) ring "R" is localized with respect to the multiplicatively closed subset "S = ∩ (R pi)", where the "pi" are finitely many prime ideals.


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