 
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
