Jacobson ring

From Wikipedia, the free encyclopedia

In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.

Jacobson rings were introduced independently by Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Goldman (1951), who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz.

Jacobson rings and the Nullstellensatz

Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of Hilbert's Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I.

In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes. For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.

Examples

  • Any field is a Jacobson ring. Any principal ideal domain or Dedekind domain with an infinite number of prime ideals is a Jacobson ring, but principal ideal domains with only a finite number of prime ideals are not Jacobson rings unless they are fields or the null ring.
  • Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, in particular the coordinate ring of any affine algebraic set, is a Jacobson ring.
  • A local ring of dimension 0 (e.g., local Artinian ring) is a Jacobson ring, but a local ring of dimension greater than 0 is not a Jacobson ring.
  • (Amitsur 1956) showed that any countablely generated algebra over an uncountable field is a Jacobson ring.

Characterizations

The following conditions on a commutative ring R are equivalent:

  • R is a Jacobson ring
  • Every prime ideal of R is an intersection of maximal ideals.
  • Every radical ideal is an intersection of maximal ideals.
  • Every Goldman ideal is maximal.
  • Every integral quotient ring of R has vanishing Jacobson radical.
  • In every quotient ring, the nilradical is equal to the Jacobson radical.
  • Every finitely generated algebra over R that is a field is finitely generated as an R-module. (Zariski's lemma)
  • Every prime ideal P of R such that R/P has an element x with (R/P)[x–1] a field is a maximal prime ideal.
  • The spectrum of R is a Jacobson space, meaning that every closed subset is the closure of the set of closed points in it.
  • (For Noetherian rings R): R has no prime ideals P such that R/P is a 1-dimensional semi-local ring.

References

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.