Editing Semi-local ring (section)

== Examples ==
* Any right or left [[Artinian ring]], any [[serial ring]], and any [[semiperfect ring]] is semi-local. 
* The quotient <math>\mathbb{Z}/m\mathbb{Z}</math> is a semi-local ring. In particular, if <math>m</math> is a prime power, then <math>\mathbb{Z}/m\mathbb{Z}</math> is a local ring.
* A finite direct sum of fields <math>\bigoplus_{i=1}^n{F_i}</math> 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''<sub>1</sub>, ..., ''m<sub>n</sub>''
:<math>R/\bigcap_{i=1}^n m_i\cong\bigoplus_{i=1}^n R/m_i\,</math>.
:(The map is the natural projection). The right hand side is a direct sum of fields.  Here we note that ∩<sub>i</sub> m<sub>i</sub>=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 [[Localization of a ring|localized]] with respect to the [[multiplicatively closed]] subset ''S = ∩ (R \ p<sub>i</sub>)'', where the ''p<sub>i</sub>'' are finitely many [[prime ideal]]s.