Math 818: Problem set 5, due February 18

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, February 18, 2005. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
  1. Let R be the ring Z/(m) and consider the ring S = R[x]/(x2-x+1).
  2. Let P be an ideal in a (nonzero commutative) ring R (with 1).
  3. A subset S of a domain R, such that S is closed under multiplication and contains 1 but does not contain 0 is said to be a multiplicative set. Let F be the field of fractions for R.
  4. Let R denote the polynomials C[x] with complex coefficients. Let f be a nonzero polynomial in R.
  5. If R and S are rings then so is RxS, where (a,b)*(c,d) = (ac,bd) and (a,b)+(c,d) = (a+c,b+d). Find a ring isomorphism f: R[x]/(x2-1) -> RxR, where R denotes the reals. Prove that your choice of f is indeed an isomorphism. [Hint: To define f, first define an appropriate homomorphism f: R[x] -> RxR.]