Page 129 - Advanced Linear Algebra

Modules I: Basic Properties 113

The following simple submodules play a special role in the theory.

Definition Let 4 be an -module. A submodule of the form
9
ºº#»» ~ 9# ~ ¸ # “  9¹

for #4 is called the cyclic submodule generated by .
#
Of course, any finite-dimensional vector space is the direct sum of cyclic
submodules, that is, one-dimensional subspaces. One of our main goals is to
show that a finitely generated module over a principal ideal domain has this
property as well.

Definition An -module 4 is said to be finitely generated if it contains a
9
finite set that generates 4 . More specifically, 4 is -generated if it has a
generating set of size although it may have a smaller generating set as
(

)
well .
Of course, a vector space is finitely generated if and only if it has a finite basis,
that is, if and only if it is finite-dimensional. For modules, life is more
complicated. The following is an example of a finitely generated module that
has a submodule that is not finitely generated.

Example 4.2 Let be the ring -´% Á% Áõ of all polynomials in infinitely
9

many variables over a field - . It will be convenient to use ? to denote
%Á %Á Ã and write a polynomial in 9 in the form ²?³. Each polynomial in(

9 ) 9, being a finite sum, involves only finitely many variables, however. Then
is an 9 -module and as such, is finitely generated by the identity element
²?³ ~ .
Now consider the submodule of all polynomials with zero constant term. This
:
module is generated by the variables themselves,
: ~ ºº%Á %Á à »»

However, is not finitely generated. To see this, suppose that . ~ ¸ Á Ã Á ¹
:
%
:
is a finite generating set for . Choose a variable that does not appear in any
of the polynomials in . Then no linear combination of the polynomials in .
.
. For if
can be equal to %

% ~ ²?³ ²?³

~