[answered] Math 113 Pierre Simon pierre.simon@berkeley. Read the chapt

Problem 2

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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.

 

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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.

 

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3. Show that if p ? R is a prime ideal containing I, then I ? p.

 

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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

 

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distinct prime numbers and ei ? Z+ . Show that nZ = J(nZ) = 1?i?n pi Z.

 

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