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.