Page 127 - Advanced Linear Algebra
P. 127
Modules I: Basic Properties 111
elements of , that is, scalar multiplication is the ring multiplication. The
9
defining properties of a ring imply that the defining properties of the -
9
module 9 are satisfied. We shall use this example many times in the
sequel.
Importance of the Base Ring
Our definition of a module requires that the ring of scalars be commutative.
9
Modules over noncommutative rings can exhibit quite a bit more unusual
behavior than modules over commutative rings. Indeed, as one would expect,
the general behavior of -modules improves as we impose more structure on
9
the base ring . If we impose the very strict structure of a field, the result is the
9
very well behaved vector space.
To illustrate, we will give an example of a module over a noncommutative ring
that has a basis of size for every integer ! As another example, if the
are linearly
base ring is an integral domain, then whenever #Á Ã Á #
9
independent over so are # Á Ã Á # for any nonzero 9 . This can fail
when is not an integral domain.
9
We will also consider the property on the base ring that all of its ideals are
9
finitely generated. In this case, any finitely generated 9 -module 4 has the
property that all of its submodules are also finitely generated. This property of
9 9-modules fails if does not have the stated property.
When is a principal ideal domain such as or - ( { ´ % µ ) , each of its ideals is
9
generated by a single element. In this case, the -modules are “reasonably” well
9
behaved. For instance, in general, a module may have a basis and yet possess a
submodule that has no basis. However, if is a principal ideal domain, this
9
cannot happen.
Nevertheless, even when is a principal ideal domain, -modules are less well
9
9
behaved than vector spaces. For example, there are modules over a principal
ideal domain that do not have any linearly independent elements. Of course,
such modules cannot have a basis.
Submodules
Many of the basic concepts that we defined for vector spaces can also be
defined for modules, although their properties are often quite different. We
begin with submodules.
Definition A submodule of an -module 4 is a nonempty subset of 4 that
:
9
is an -module in its own right, under the operations obtained by restricting the
9
operations of 4 to . We write : 4 to denote the fact that is a submodule
:
:
of 4 .
