Name:
Problem Set 1
Math 416, Section 500, Spring 2014
Due: Thursday, January 23.
Review Sections 26, 27 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Thursday, January 23. Remember to fully justify all your answers,
and provide complete details.
1. Carefully read the entire course website. Send an email to your instructor containing:
• Math416 in the subject line;
• a review of the course website (What did you like? What would you change? List any
typos you discovered, etc.); and
• an acknowledgement that you understand the policies and procedures for this course.
2. Find all ideals N of Z20 . In each case, compute Z20 /N, that is, find a known ring isomorphic to
the quotient ring.
3. Exercise 19, p. 244.
4. Show that if N1 and N2 are ideals of R, then N1 ∩ N2 is also an ideal of R.
5. Let R be a commutative ring and N an ideal of R. Define the radical of N to be
√
N B {x ∈ R | ∃n ∈ Z+ such that xn ∈ N}.
√
Show that N is an ideal of R.
6. Exercise 37, p. 245.
7. Find all prime and maximal ideals of Z20 .
8.
a. Give an example (in a commutative ring with unity) of a prime ideal that is not maximal.
b. Let F be a field. Show that every proper nontrivial prime ideal of F[x] is maximal.
9. Exercise 34, p. 254.
10. Exercise 35, p. 254
11. Exercise 36, p. 254.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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