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 )