Math 724

Fall 2001

Homework 2

 

Due October 8, 2001.

 

1.      Show that a the following conditions on a ring are equivalent:

a)   Every ideal of  is finitely generated.

b)   Every prime ideal of  is finitely generated.

c)    satisfies the ascending chain condition on ideals.

 

2.      A ring,, is said to be Von-Neumann regular if for all  there exists a  such that  Show that if  is Von-Neumann regular, then  for all maximal ideals (Actually these two conditions are equivalent).

 

 

3.      Let  be a Noetherian ring,  an ideal and  a multiplicatively closed set. Establish the following:

a)  is Noetherian.

b)  is Noetherian.

c) If  is a one-dimensional domain (that is, all nonzero primes are maximal) and

then  is Artinian.

 

4.      Show that any overring of the integers is a localization. Give an example of an overring of a one-dimensional domain that is not a localization.