Math 721

Spring 2001

Homework 3

1. Let be a ring with identity. Show that the following conditions are equivalent:

a) Every (unitary) module is projective.

b) Every (unitary) module is injective.

c) Every short exact sequence of modules is split exact.

2. Let be a divisible abelian group.

a) Show that every homomorphic image of is also divisible.

b) Show that any abelian group may be embedded in a divisible abelian group.

3. Let and denote families of unitary modules. Prove the following:

a) is projective if and only if is projective for all

b) is injective if and only if is injective for all

4. Let be a field and be a free module.

a) Show that is a ring.

b) Show that for any positive integer (That is, the module, has a basis of any finite cardinality, hence does not have the invariant dimension property).