Semi-local ring

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.

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