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.