Math 420/620
Fall 1999
Exam 2
(You may assume that all rings are commutative with identity.)
- Consider the ring of rational integers Z .
- Prove that Z is a principal ideal domain (PID).
- Show that the polynomial ring Z[x] is not a PID.
- Let
be a homomorphism of commutative rings (you may assume that ![](Image617.gif)
).
- Show that if
is an ideal, then
is an ideal of ![](Image620.gif)
- Show that if
is prime, then so is ![](Image622.gif)
- Let
be a family of prime ideals in ![](Image624.gif)
- Show that
is a radical ideal of ![](Image626.gif)
- (G)
Show that if
is a radical ideal, then ![](Image628.gif)
.
- The nilradical of
is the collection of all nilpotent elements of ![](Image629.gif)
.
- Show that the nilradical is an ideal of
![](Image630.gif)
- (G)
Show that the nilradical is the intersection of all prime ideals of ![](Image631.gif)
- The Jacobson radical (
) of
is the intersection of all maximal ideals of ![](Image633.gif)
- Show that the following conditions are equivalent:
.
is a unit of
for all ![](Image636.gif)
- Show that the nilradical is always contained in the Jacobson radical.
- Compute the nilradical and the Jacobson radical for the rings Z, Z12.
- (G)
Show that if ![](Image600.gif)
is finite, then the Jacobson radical and the nilradical are the same.