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).

 

 

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

 

  1. 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.