### 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).
• (a) Show that every element of S is the image (under the quotient homomorphism) of an element of the form ax + b in R[x], for some a and b in R.
• (b) In each case (i.e., m=2 and m=3), determine the product (x+1)(x-1) in S (i.e., write the product as ax+b for some a and b), and find the inverse of x-1 if it exists.
• (c) Determine all ideals of S, if m = 2 (explain). Is S a field? Why or why not?
• (d) Determine all ideals of S, if m = 3 (explain). Is S a field? Why or why not?
2. Let P be an ideal in a (nonzero commutative) ring R (with 1).
• (a) A proper ideal with either of the following three properties is called a prime ideal. Show that the properties are equivalent:
• (i) If I and J are ideals in R and IJ is contained in P, then either I is contained in P or J is contained in P.
• (ii) If a and b are in R with ab in P, then either a is in P or b is in P.
• (iii) The quotient R/P is a domain.
• (b) Show that a maximal ideal is prime.
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.
• (a) If P is a prime ideal, prove that S = R-P is a multiplicative set.
• (b) If r is a nonzero element of R, prove that S = {1, r, r2, ...} is a multiplicative set.
• (c) Let S be a multiplicative set. The subset of F of all elements of F of the form a/s where s is in S, forms a subring of F denoted S-1R. (You may assume this, and that R is a subring of S-1R. In the case that S= R-P for a prime ideal P, the ring S-1R is often denoted RP. In the case that S = {1, r, r2, ...} for some nonzero element r of R, the ring S-1R is often denoted Rr.)
• (i) Find S-1R in case R = Z, F = Q, and S = Z - (2).
• (ii) Find S-1R in case R = Z, F = Q, and S = {1, 2, 22, ...}. (Elements of S-1R are called dyadic rationals.)
4. Let R denote the polynomials C[x] with complex coefficients. Let f be a nonzero polynomial in R.
• (a) Let I be an ideal in Rf, and let J be the intersection of I with the subring R. Show that J is an ideal, hence principal and so of the form J = gR for some g in R; show that I = gRf.
• (b) Let h be in R. Show that h is invertible in Rf if and only if every root of h is a root of f. If h = x-c for some complex number c, show that hRf is a maximal ideal if and only if f(c) is not 0.
• (c) Show that there is a bijection between the points c of the complex plane C such that f(c) is not 0, and maximal ideals in Rf. (This explains the terminology "localizing" for adding invertible elements. If you localize by inverting a polynomial f with more roots, the maximal ideals of Rf correspond to a smaller open subset of the complex plane C.)
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.]