G domain

In mathematics, an integral domain D is a G-domain if and only if:

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

A G-ideal is defined as an ideal I ⊂ R such that 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 maximal ideals are G-ideals. This leads to a simplified proof of the Nullstellensatz.

It is known that given T ⊃ R , a ring extension of a G-domain, T is algebraic over R if and only if every ring extension between T and 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).