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.
 Do Problem #12, p. 380.
 Do Problem #10, p. 381.
 Do Problem #24, p. 381.
 Let R be a perhaps noncommutative ring.
If r^{2} = r for all r in R, prove that R is
in fact commutative.
 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.