Goldman domain

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

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

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., 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