HW #3 due Fri Feb 2 -- corrected

1a. Let f:R->S be a ring homomorphism, and M a left S-module. Show that M has also the structure of an R-module.

b. Even better... show that there is a functor S-Mod -> R-Mod (the categories of left modules), where the above says what to do with the objects.

c. Even better... show that there is a functor Ring^op -> Cat, taking R |-> R-Mod.

2. Let M be a left R-module, and A,B two submodules. Show that the abelian subgroups A+B, A intersect B are also submodules.

3a. Let b be an element of the ring R. Show that the smallest 2-sided ideal containing b is RbR := {sum_i r_ibs_i}.

b. Let R = M_n(C), nxn complex matrices, and b a nonzero matrix. Show that RbR = R, i.e. this R is a "simple" ring.

4. Let A,B be two abelian groups. Show that Hom(A,B) is not only an abelian group, but a left End(A)^op x End(B) module End(B)-module, and also a left End(A)^op module.
(We will later show it to be a left End(A)^op tensor End(B) module.)

5. Recall the definitions of initial (resp. terminal) object X in a category; there exists a unique morphism out of X to (resp. into X from) any other object Y.

Find an initial and a terminal object in the category Ring = {rings with units, s.t. homomorphisms preserve the units}. (As the example Set shows, they need not be the same object.)