semilocal ring

In mathematics, a semi-local ring is a ring for which ''R''/J(''R'') is a semisimple ring, where J(''R'') is the Jacobson radical of ''R''.

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 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 (right/left/two-sided) ideal.

Examples

* Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
* The quotient $\mathbb/m\mathbb$ is a semi-local ring. In particular, if $m$ is a prime power, then $\mathbb/m\mathbb$ is a local ring.
* A finite direct sum of fields $\bigoplus_^n$ 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''
:$R/\bigcap_^n m_i\cong\bigoplus_^n R/m_i\,$.
:(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩_i m_i=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 localized with respect to the multiplicatively closed subset ''S = ∩ (R \ p_i)'', where the ''p_i'' are finitely many prime ideals.

Textbooks

*
*

Ring theory
Localization (mathematics)

{{Abstract-algebra-stub