Math 724

Fall 2001

Homework 4

 

  1. Let  be a Dedekind domain. Show that any ideal of  may be generated by at most two elements.

 

  1. Let  be one-dimensional and Noetherian.

a)      Show that the integral closure of  is Dedekind.

b)      Show that if  is local, then the integral closure of  is a PID with only finitely many primes.

 

  1. Let  be a domain and  a prime ideal such that  Show that  is a Noetherian valuation domain.

 

  1. Give an example of a domain  with quotient field  and an overring  with  or show that no such example can exist.