Goldman domain

In mathematics, a Goldman domain or G-domain is an integral domain "A" whose field of fractions is a finitely generated algebra over "A". 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.
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".
Alternative definition.
An integral domain formula_1 is a G-domain if and only if:
A G-ideal is defined as an ideal formula_8 such that formula_9 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.
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 G-ideals are maximal ideals. This leads to a simplified proof of the Nullstellensatz.
It is known that given formula_10, a ring extension of a G-domain, formula_11 is algebraic over formula_12 if and only if every ring extension between formula_11 and formula_12 is a G-domain.
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). 