Summary: Rings and Algebras Problem set #7: Solutions Oct. 27, 2011.
1. Describe the indecomposable injective Abelian groups.
Solution. An indecomposable Abelian group is the injective envelope of any of its subgroups, in particular
the envelope of any of its cyclic subgroups. Moreover if a group has a nontrivial decomposition then so does its
injective envelope. Thus it is enough to find the injective envelope for Zp for p prime and for Z. It is easy to
see that E(Zp ) = Zp # and E(Z) = Q, since the latter modules are clearly divisible (hence injective) and they
contain the corresponding cyclic subgroup as an essential subgroup.
2. Find the indecomposable decomposition of the injective envelope of the following abelian
groups: Z +
15 , Z +
100 , Z[i], C × .
Solution. E(Z +
15 ) = Z3 # # Z5 # , E(Z +
100 ) = Z2 # # Z5 # , E(Z[i]) = Q #Q, E(C × ) = 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.