Math 421/621

Spring 2000

Homework 3

1. Suppose M and N are R-modules and there exist R-module homomorphisms: and such that . Show that .

2. Show that the following conditions are equivalent:

a) P is a projective R-module.

b) Every short exact sequence of the form is split exact.

3. Show that the following conditions are equivalent:

a) I is an injective R-module.

b) Every short exact sequence of the form is split exact.

4. (G) We are given a short exact sequence of R-module homomorphisms:

.

Show that the following conditions are equivalent:

a) There is an R-module homomorphism such that .

b) There is an R-module homomorphism such that .

c) .

5. (G) Let be an R-module homomorphism such that . Show that .

6. We showed in class that any projective module is the summand of a free module. Show the converse, that is, show that if P is the summand of a free module (i.e. there is a module K and a free module F such that ) then P is projective.