Rings and Algebras Problem set #7. Oct. 27, 2011. 1. Describe the indecomposable injective Abelian groups. 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 0 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 Collections: Mathematics