Math 420/620

Fall 1999

Exam 2

 

(You may assume that all rings are commutative with identity.)

 

  1. Consider the ring of rational integers Z .
  1. Prove that Z is a principal ideal domain (PID).
  2. Show that the polynomial ring Z[x] is not a PID.

 

  1. Let be a homomorphism of commutative rings (you may assume that
).
  1. Show that if is an ideal, then is an ideal of
  2. Show that if is prime, then so is

 

  1. Let be a family of prime ideals in
  1. Show that is a radical ideal of
  2. (G) Show that if is a radical ideal, then
.

 

  1. The nilradical of is the collection of all nilpotent elements of
.
  1. Show that the nilradical is an ideal of
  2. (G) Show that the nilradical is the intersection of all prime ideals of

 

 

  1. The Jacobson radical () of is the intersection of all maximal ideals of
  1. Show that the following conditions are equivalent:
  1. .
  2. is a unit of for all

 

  1. Show that the nilradical is always contained in the Jacobson radical.
  2. Compute the nilradical and the Jacobson radical for the rings Z, Z12.
  3. (G) Show that if
is finite, then the Jacobson radical and the nilradical are the same.