Math 793

Summer 2001

Homework 6

Turn in at least two by the last day of class, August 3.

1. Let be a valuation domain (that is, is a domain where given any two nonzero elements either divides or divides ). Show that the following statements are equivalent:

a) is atomic.

b) is a UFD.

c) is a PID.

(So valuation domains are either “really good” or “really bad” from a factorization point of view).

2. Give an example of a one dimensional valuation domain that is not atomic.

3. Consider the rings

a) Show that is a UFD.

b) Explicitly show that is not an HFD.

c) Does the bad factorization that you produced in part b) show that is also not an HFD? If not, then show why.