Math 724

Fall 2001

Final Exam

1. A domain with quotient field is said to be root closed if for all ,

a) Translate the property root closed to the group of divisibility (e.g. is root closed if and only if is …).

b) Translate the property completely integrally closed to the group of divisibility as per part a).

2. Let denote a finite family of partially ordered abelian groups. We consider and declare if and only if (This is called the product order).

a) Show that is a partially ordered abelian group.

b) Construct an integral domain with group of divisibility isomorphic to under the product order (where denotes the integers).

3. Let be the integral closure of the integers in the complex numbers.

a) Show that the group of divisibility of has no minimal positive elements.

b) Construct a valuation domain with the property that for any nonzero prime ideal there is a nonzero prime with

c) Construct a valuation domain, with the property that the group of divisibility of has no minimal positive elements, but every proper localization of (that is not a field) has a group of divisibility with minimal positive elements.