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 R-module M we have R/I 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 R-module M we have R/I R M M/IM as Abelian groups. b) Compute Zm Z Zn. 2. Show that for any R-module M the functors HomR(M, -) and HomR(-, M) are left exact. 3. Show (by direct computation that for any R-module 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 (Ai, i,j | i j), where Ai are objects in C and i,j HomC(Ai, Aj) 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 Ai of a direct system is an object A together with morphisms i HomC(Ai, 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 HomC(Ai, B)), satisfying similar commuting relations, we have a unique Collections: Mathematics