Math 721

Spring 2001

Homework 1

These problems will be due shortly after I arrive back. Please begin reading the sections in Hungerford’s book on modules and attempt to answer them.

1. Given an module homomorphism show that

2. Let be an module homomorphism such that Show that

3. Let be an module with a commutative ring.

a) If then show that is an ideal of (If then we say that is a torsion element of )

b) If is an integral domain and is the set of torsion elements of , show that is a submodule of

c) Is b) true if “integral domain” is omitted?

4. Let be a ring and consider as a module over itself. Is it true that an module homomorphism is also a homomorphism of rings?

5. Give an example of a finitely generated module and give an example of an module that cannot be finitely generated.