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firstname.lastname@example.org Homework 10
due November 22
Reading assignment: Read the chapters on euclidean rings and PID.
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.
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
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) = 1?i?n pi Z.
This question was answered on: Sep 18, 2020
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