### Math 818: Problem set 8,
due March 25

*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, March 25, 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.)

Assume all rings are commutative with 1 not equal to 0
unless specifically stated otherwise.

- The
*annihilator* of an R-module V is the set ann(V)
={r in R : rv=0 for all v in V}.
- (a) Prove that ann(V) is an ideal of R.
- (b) Find the annihilator of the direct product
**R** x **R**, where the **R**[x]-module structure of
**R** x **R** is given by the following rule for
multiplication by elements of **R**[x]: f(x)(a,b) = (f(1)a,f(0)b).

- Let I be an ideal in a ring R.
- (a) Prove that I is a free R-module if and only if
I=(r) for some r in R which is not a nonzero zero divisor.
- (b) Apply (a) to conclude that the ideal (x,y) in
**R**[x,y]
is not a free **R**[x,y]-module.

- Show each of the following statements is false by
giving an explicit counterexample for each one.
- (a) Every set of generators for a free module of finite rank
contains a basis.
- (b) Every independent set in a free module of finite rank
is contained in a basis.
- (c) Let A and B be 2x2 matrices with entries in the
ring noncommutative
ring M
_{2}(**Z**) of 2x2 integer matrices.
Thus each entry of A and of B is itself a 2x2 matrix.
Then det(AB) = det(A)det(B).

- If R is a ring such that every finitely generated
R-module is free, prove that R is a field.
- Do problem 6 on p. 485.
- Do problem 10 at the top of p. 486.