Semi-local Ring - Examples

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.

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