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
100, Z[i], C×
3. Take the algebra A =
and consider the right A-modules S1 = ( K 0 ), S2 =
( 0 K ) and P1 = ( 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 S1, S2 are simple but P1 is not.
b) Two of the modules listed above are injective. Which are the injective ones?
4. An R-module M is divisible if M = rM for every non-zero-divisor 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.