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.