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 )