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 .

 

  1. 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.

 

  1. 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.

 

  1. (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)      .

 

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

 

  1. 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.