Semi-local ring

From Wikipedia, the free encyclopedia

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.

[edit] Examples

A finite direct sum of fields $\bigoplus_{i=1}^n{F_i}$ 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, ..., m_n

R / ∩_i m_i ≅ ⊕_i R / m_i.

(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 \ p_i), where the p_i are finitely many prime ideals.