Goldman domain

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In mathematics, a Goldman 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.

Notes[edit]

  1. ^ Goldman domains/ideals are called G-domains/ideals in (Kaplansky 1974).
  2. ^ Kaplansky, pp. 13

References[edit]