View source for Semi-local ring

{{Short description|Algebraic ring classification}}
{{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 (mathematics)|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 ideal]]s".

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 <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.

==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)]]


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