Math 421/621
Spring 2000
Homework 3
- Suppose M and N are
R-modules and there exist R-module homomorphisms: and such that . Show that .
- 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.
- 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.
- (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)
.
- (G) Let be an R-module homomorphism such that . Show that .
- 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.