Summary: Rings and Algebras Problem set #7. Oct. 27, 2011.
1. Describe the indecomposable injective Abelian groups.
2. Find the indecomposable decomposition of the injective envelope of the following abelian
groups: Z +
15 , Z +
100 , Z[i], C × .
3. Take the algebra A = # K K
# and consider the right Amodules S 1 = ( K 0 ), S 2 =
( 0 K ) and P 1 = ( K K ) with the obvious module structure (i. e. ``make'' some elements of
the matrix product 0 in order to have a module).
a) Show that S 1 , S 2 are simple but P 1 is not.
b) Two of the modules listed above are injective. Which are the injective ones?
4. An Rmodule M is divisible if M = rM for every nonzerodivisor r # R.
a) Show that if E is injective then it is divisible.
b) Assume that R is a principal ideal domain. Then E is injective if and only if it is divisible.
5. a) Let R be principal ideal domain with quotient field Q. Show that Q is the injective
envelope of RR.
b) Show the same for arbitrary commutative domains.
6. An Rmodule C is called a cogenerator if every Rmodule can be embedded into a direct