Editing Semi-local ring (section)

{{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.