Semi-local ring

In mathematics, a semi-local ring is a ring for which R/J is a semisimple ring, where J is the Jacobson radical of R.
The above definition is satisfied if R has a finite number of maximal right 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 ideals".
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 ideal.

Examples

Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
The quotient is a semi-local ring. In particular, if is a prime power, then is a local ring.
A finite direct sum of fields 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 m₁,..., m_n
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 ring R is localized with respect to the multiplicatively closed subset S = ∩ , where the p_i are finitely many prime ideals.