101100/Homework Assignment 4

This assignment is due at class time on Tuesday, November 30, 2010.
Solve the following questions
Problem 1. Prove that a ring is a PID iff it is a UFD in which for every nonzero .
Problem 2. (Selick) In a ring , and element is called "nilpotent" if for some positive , . Let be the set of all nilpotent elements of .
 Prove that if is commutative then is an ideal.
 Give an example of a noncommutative ring in which is not an ideal.
Problem 3. (comprehensive exam, 2009) Let be a commutative ring. Show that a polynomial is invertible in iff its constant term is invertible in and the rest of its coefficients are nilpotent.
Problem 4. (Lang) Show that the ring is a PID and hence a UFD. What are the units of that ring?
Problem 5. (Dummit and Foote) In , find the greatest common divisor of and , and express it as a linear combination of these two elements.
Problem 6. (Withdrawn, do not submit) Show that the quotient ring is not a UFD.