Math 726
Fall 2002
Exam 1
Due
Monday, October 14, 2002.
1.
Prove
that if are
modules (
is commutative with identity), then there is an
module isomorphism:
2.
Give
an example where is not isomorphic to
and an example where
is not isomorphic to
3.
Give
an example where is not isomorphic to
and an example where
is not isomorphic to
4.
Give
an example to show that tensoring does not commute with direct product in
general (this is one of the reasons that, for example, power series are
sometimes difficult to work with).
5.
Show
that (in either variable) the functor is not necessarily an exact functor.
6.
Let
be an additive
functor from the category of
modules to the category of abelian groups. Show that if
is a split exact sequence of module homomorphisms, then
is a summand of
Use this to show that
preserves finite
sums.