Page 159 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 143

Á Á Á à Á ³  : for some  9¹

0 ~ ¸  9 “ ² Á à Á c Á à Á c

0
Then it is easy to see that is an ideal of and so 9 0 ~ º » for some  9 .
Let
" ~ ² ÁÃÁ c Á Á ÁÃÁ ³  :

We claim that
8 ~¸" “ ~ Á à Á and £ ¹

is a basis for . As to linear independence, suppose that
:
8 ~¸" Á Ã Á " ¹

and that
" bÄb " ~

Then comparing the th coordinates gives ~ and since £ , it

8
follows that ~ . In a similar way, all coefficients are and so is linearly

independent.
8
To see that spans , we partition the elements  % : according to the largest
:
coordinate index ²%³ with nonzero entry and induct on ²%³ . If ²%³ ~ , then
%~ , which is in the span of . Suppose that all % : with ²%³  are in
8
8
the span of and let ²%³ ~ , that is,
% ~ ² ÁÃÁ Á ÁÃÁ ³

where £ . Then 0 and so £ and ~ for some 9 .

8
Hence, ²% c " ³  and so & ~ % c "  ºº »» and therefore %  ºº »» .
8

Thus, is a basis for .
8
:
The previous proof can be generalized in a more or less direct way to modules
is the -module
of arbitrary rank. In this case, we may assume that 4~ ²9 ³ 9
of functions with finite support from to , where is a cardinal number. We

9
use the fact that is a well-ordered set, that is, is a totally ordered set in which

any nonempty subset has a smallest element. If  , the closed interval ´ Á µ
is

´ Á µ ~¸% “ % ¹

Let : 4 . For each   , let
4 ~ ¸  : “ supp ² ³ ‹ ´ Á µ¹

Then the set
0~ ¸ ² ³ “  4 ¹