### Math 818: Problem set 4, due February 11

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 11, 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.)
• (a) Do Problem #4, p. 379.
• (b) Do Problem #6, p. 379.
1. Do Problem #12, p. 380.
2. Do Problem #10, p. 381.
3. Do Problem #24, p. 381.
4. Let R be a perhaps noncommutative ring. If r2 = r for all r in R, prove that R is in fact commutative.
5. Let I and J be ideals of a ring R, such that I + J = R.
• (a) Prove that the intersection of I and J is equal to IJ. Give an example of this in case R = Z.
• (b) Show that R/IJ is isomorphic to R/I x R/J. (This is one version of the Chinese Remainder Theorem.) Give an example of this in case R = Z.