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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
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