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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 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, ..., mn'' :. :(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=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 \ pi)'', where the ''pi'' are finitely many prime ideals. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「semilocal ring」の詳細全文を読む スポンサード リンク
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