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.[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.

Alternative definition

An integral domain ${\displaystyle D}$ is a G-domain if and only if:

1. Its quotient field is a simple extension of ${\displaystyle D}$
2. Its quotient field is a finite extension of ${\displaystyle D}$ (Note this would mean the quotient field is integral over D and thus D has Krull dimension zero; i.e., a field.)
3. Intersection of its nonzero prime ideals (not to be confused with nilradical) is nonzero
4. There is a non-zero element ${\displaystyle u}$ such that for any nonzero ideal ${\displaystyle I}$, ${\displaystyle u^{n}\in I}$ for some ${\displaystyle n}$.[3]

A G-ideal is defined as an ideal ${\displaystyle I\subset R}$ such that ${\displaystyle R/I}$ 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.[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 G-ideals are maximal ideals. This leads to a simplified proof of the Nullstellensatz.[5]

It is known that given ${\displaystyle T\supset R}$, a ring extension of a G-domain, ${\displaystyle T}$ is algebraic over ${\displaystyle R}$ if and only if every ring extension between ${\displaystyle T}$ and ${\displaystyle R}$ 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

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

• Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, ISBN 0-226-42454-5, MR 0345945
• Picavet, Gabriel (1999), "About GCD domains", in Dobbs, David E. (ed.), Advances in commutative ring theory. Proceedings of the 3rd international conference, Fez, Morocco, Lect. Notes Pure Appl. Math., 205, New York, NY: Marcel Dekker, pp. 501–519, ISBN 0824771478, Zbl 0982.13012