Math 726

Fall 2002

Homework 1

Due Monday, September 9, 2002. Do not forget to mark your “favorite” problem.

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. Consider the exact sequences:

and

a) Show that the sequence

is also exact.

b) Use part a) to show that any exact sequence can be obtained by “splicing” together suitable short exact sequences.

5. (The 3x3 Lemma) Consider the following commutative diagram:

a) Show that if the columns are exact and the bottom two rows are exact, then so is the top.

b) Show that if the columns are exact and the top two rows are exact, then so is the bottom.