Rings and Algebras Problem set #9. Nov. 17, 2011. 1. a) Let I be a right ideal in a ring R. Show that for any Rmodule M we have Summary: Rings and Algebras Problem set #9. Nov. 17, 2011. 1. a) Let I be a right ideal in a ring R. Show that for any R­module M we have R/I# R M # M/IM as Abelian groups. b) Compute Zm# Z Z n . 2. Show that for any R­module M the functors HomR (M,-) and HomR (-, M) are left exact. 3. Show (by direct computation that for any R­module M the functor -# R M is right exact. 4. A partially ordered set (I, #) is called a directed set if each pair of elements has an upper bound. For a given directed set (I, #) and a category C, a system (A i , # i,j | i # j), where A i are objects in C and # i,j # Hom C (A i , A j ) is called a direct system if # i,i is the identity morphism and # i,k = # j,k # # i,j for each i # j # k. The direct limit lim # A i of a direct system is an object A together with morphisms # i # Hom C (A i , A) so that # i = # j # # i,j for each i # j, furthermore it is universal in the sense that for any other system (B, # i # Hom C (A i , B)), satisfying similar commuting relations, we have a unique Collections: Mathematics