3Modules
IB Groups, Rings and Modules
3 Modules
Finally, we are going to look at modules. Recall that to define a vector space,
we first pick some base field
F
. We then defined a vector space to be an abelian
group
V
with an action of
F
on
V
(i.e. scalar multiplication) that is compatible
with the multiplicative and additive structure of F.
In the definition, we did not at all mention division in
F
. So in fact we can
make the same definition, but allow
F
to be a ring instead of a field. We call
these modules. Unfortunately, most results we prove about vector spaces do use
the fact that
F
is a field. So many linear algebra results do not apply to modules,
and modules have much richer structures.