Math 421/621

Spring 2000

Homework 2

1. Show that, in general, a maximal linearly independent subset of a free R-module need not be a basis.

2. Let R be a ring, show that as left R-modules

3. Let R be a commutative ring with 1. Show that the following conditions are equivalent:

a) Every R-module is free.

b) R is a field.

4. An abelian group is called torsion is every element is of finite order. Let A be a torsion abelian group, compute .

5. Let R be a commutative ring with 1, and let M be an R-module. Compute .

6. Show that any submodule, M, of the free Z-module is free and that .