Math 113 Pierre Simon pierre.simon@berkeley.edu Homework 10 due November 22 Reading assignment: Read the chapters on euclidean rings and PID. Problem 1 Let R be a principal ideal domain (PID) and let a, b, c R be non-zero. Let d be a greater common divisor of a and b, and write a = a0 d and b = b0 d. We want to solve the equation ax + by = c. 1. Show that there exists x, y R such that ax + by = c if and only if d|c. 2. Assume that for some x0 , y 0 R, we have ax0 = by 0 . Show that there is u R such that x0 = ub0 and y 0 = ua0 . 3. Assume now that x0 , y0 R are such that ax0 + by0 = c. Show that x, y R are such that ax + by = c if and only if there exists u R such that x = x0 + ub0 and y = y0 ua0 . 4. Let P = X 3 7X + 6 and Q = 2X 2 + 5X 3. Execute the euclidean algorithm for P and Q. 5. Let R = X 2 9 Q[X]. Find S and T Q[X] such that P S + QT = R. Problem 2 Let R be a commutative ring with identity 1 6= 0. Let I R be an ideal. 1. Define J(I) to be the intersection of all maximal ideals J R containing I. Show that J(I) is an ideal of R. 2. Let I be the set of x R for which there is n Z+ with xn I. Show that I is an ideal of R. 3. Show that if p R is a prime ideal containing I, then I p. 4. Deduce that I J(I). 5. Assume that n Z+ , n > 1 has the prime decomposition n = pe11Q penn , where the pi 's are pairwise distinct prime numbers and ei Z+ . Show that nZ = J(nZ) = 1in pi Z