Semi-local ring

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For the older meaning of a Noetherian ring with a topology defined by an ideal in the Jacobson radical, see Zariski 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. (Lam 2001, §20)(Mikhalev 2002, C.7)

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[edit]

  • 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.

Textbooks[edit]