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
).
- Show that if is an ideal, then is an ideal of
- Show that if is prime, then so is
- Let be a family of prime ideals in
- Show that is a radical ideal of
- (G)
Show that if is a radical ideal, then
.
- The nilradical of is the collection of all nilpotent elements of
.
- Show that the nilradical is an ideal of
- (G)
Show that the nilradical is the intersection of all prime ideals of
- The Jacobson radical () of is the intersection of all maximal ideals of
- Show that the following conditions are equivalent:
- .
- is a unit of for all
- 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
is finite, then the Jacobson radical and the nilradical are the same.