Math 726

Fall 2002

Final Exam

Due as soon as you can, but by December 20, 2002 anyway.

1. Let and be abelian groups. Show that for all and (Hint: you may use the fact that any subgroup of a free abelian group is a free abelian group.)

2. Given the exact sequence:

show that if and are flat, then is flat. Is the converse true?

3. Show that if is any abelian group, then

4. Let R be an integral domain. Show that if is a torsion module then is torsion for all and for all (This is an important step in showing the result that is torsion for all and )