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 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. Collections: Mathematics