Goldman domain

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In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A.[1] They are named after Oscar Goldman.

An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain. There exists a Goldman domain where all nonzero prime ideals are maximal although there are infinitely many prime ideals.[2]

An ideal I in a commutative ring A is called a Goldman ideal if the quotient A/I is a Goldman domain. A Goldman ideal is thus prime, but not necessarily maximal. In fact, a commutative ring is a Jacobson ring if and only if every Goldman ideal in it is maximal.

The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal: the radical of an ideal I is the intersection of all Goldman ideals containing I.

Formal definition[edit]

An integral domain is a G-domain if and only if:

  1. Its quotient field is a simple extension of
  2. Its quotient field is a finite extension of
  3. Intersection of its nonzero prime ideals (not to be confused with nilradical) is nonzero
  4. There is an element such that for any nonzero ideal , for some .[3]

A G-ideal is defined as an ideal such that is a G-domain. Since a factor ring is an integral domain if and only if the ring is factored by a prime ideal, every G-ideal is also a prime ideal. G-ideals can be used as a refined collection of prime ideals in the following sense: Radical can be characterized as the intersection of all prime ideals containing the ideal, and in fact we still get the radical even if we take the intersection over the G-ideals.[4]

Every maximal ideal is a G-ideal, since quotient by maximal ideal is a field, and a field is trivially a G-domain. Therefore, maximal ideals are G-ideals, and G-ideals are prime ideals. G-ideals are the only maximal ideals in Jacobson ring, and in fact this is an equivalent characterization of a Jacobson ring: a ring is a Jacobson ring when all maximal ideals are G-ideals. This leads to a simplified proof of the Nullstellensatz.[5]

It is known that given , a ring extension of a G-domain, is algebraic over if and only if every ring extension between and is a G-domain.[6]

A Noetherian domain is a G-domain iff its rank is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).[7]

Notes[edit]

  1. ^ Goldman domains/ideals are called G-domains/ideals in (Kaplansky 1974).
  2. ^ Kaplansky, p. 13
  3. ^ Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, pp. 12, 13.
  4. ^ Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, pp. 16, 17.
  5. ^ Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, p. 19.
  6. ^ Dobbs, David. "G-Domain Pairs". Trends in Commutative Algebra Research, Nova Science Publishers, 2003, pp. 71–75.
  7. ^ Kaplansky, Irving. Commutative Algebra. Polygonal Publishing House, 1974, p. 19.

References[edit]