11-1100/Homework Assignment 4

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This assignment is due at class time on Tuesday, November 29, 2011.

Solve the following questions

Problem 1. Prove that a ring R is a PID iff it is a UFD in which \gcd(a,b)\in\langle a, b\rangle for every non-zero a,b\in R.

Problem 2. (Selick) In a ring R, and element x is called "nilpotent" if for some positive n, x^n=0. Let \eta(R) be the set of all nilpotent elements of R.

  1. Prove that if R is commutative then \eta(R) is an ideal.
  2. Give an example of a non-commutative ring R in which \eta(R) is not an ideal.

Problem 3. (comprehensive exam, 2009) Let A be a commutative ring. Show that a polynomial f\in A[x] is invertible in A[x] iff its constant term is invertible in A and the rest of its coefficients are nilpotent.

Problem 4. (Lang) Show that the ring {\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C} is a PID and hence a UFD. What are the units of that ring?

Problem 5. (Dummit and Foote) In {\mathbb Z}[i], find the greatest common divisor of 85 and 1+13i, and express it as a linear combination of these two elements.

Problem 6. (Hard!) Show that the quotient ring {\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle is not a UFD.