Problem 3. Let A be any Z-module and let a ? A, n ? Z. We attempt to define the map
?<sub>a</sub>: Z/nZ ? A
by
?<sub>a</sub>(k) = ka.
Prove that ?<sub>a</sub> is a well-defined Z-module homomorphism if and only if na = 0 in A.
Use this to conclude that
Hom<sub>Z</sub>(Z/nZ, A) ? Ann<sub>A</sub>((n))
where Hom<sub>Z</sub>(Z/nZ, A) is the set of Z-module homomorphisms (i.e. the set of ?<sub>a</sub> which are
Z-module homomorphisms) and Ann<sub>A</sub>((n)) is the submodule of A which is the annihilator
of (n) ? Z (i.e. the subset of A for which na = 0).
NB! This problem looks like it's asking for a lot, but it's really checking that the map is
well-defined. Its main purpose is to show that the annihilator isn't just a made-up homework
problem, but something which gives insight into maps of R-modules.
