G-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. 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. An integral domain D {displaystyle D} is a G-domain if and only if: A G-ideal is defined as an ideal I ⊂ R {displaystyle Isubset R} such that R / I {displaystyle R/I} 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 T ⊃ R {displaystyle Tsupset R} , a ring extension of a G-domain, T {displaystyle T} is algebraic over R {displaystyle R} if and only if every ring extension between T {displaystyle T} and R {displaystyle R} 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).