# 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

## 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