Page 126 - Advanced Linear Algebra

110 Advanced Linear Algebra

together with two operations. The first operation, called addition and denoted
by b , assigns to each pair ² " Á # ³  4 d 4 , an element " b #  4 . The
second operation, denoted by juxtaposition, assigns to each pair
² Á #³  9 d 4, an element #  4. Furthermore, the following properties
must hold:
)
1 4 is an abelian group under addition.
)
2 For all Á  9 and "Á #  4
²"b#³ ~ "b #
² b ³" ~ " b "
² ³" ~ ² "³
" ~ "
The ring is called the base ring of 4 .
9
Note that vector spaces are just special types of modules: a vector space is a
module over a field.

When we turn in a later chapter to the study of the structure of a linear
transformation ²= ³ , we will think of as having the structure of a vector
=

B
space over as well as a module over ´ - % µ and we will use the notation . Put
-
=
is an abelian group under addition, with two scalar
another way, =
multiplications—one whose scalars are elements of and one whose scalars are
-
polynomials over . This viewpoint will be of tremendous benefit for the study
-
of . For now, we concentrate only on modules.

Example 4.1
1 If is a ring, the set 9 of all ordered -tuples whose components lie in 9
)
9

is an 9 -module, with addition and scalar multiplication defined
(
componentwise just as in - , )
² ÁÃÁ ³ b ² ÁÃÁ ³ ~ ² b ÁÃÁ b ³

and
² Á Ã Á ³ ~ ² Á Ã Á ³

{
for , Á  9 . For example, { is the -module of all ordered -tuples
of integers.
) 9 ² 9 ³ of all matrices of size d is an -
9
2 If is a ring, the set C Á
module, under the usual operations of matrix addition and scalar
multiplication over . Since is a ring, we can also take the product of
9
9
matrices in C Á ²9³ . One important example is 9 ~ -´%µ , whence
C Á ²-´%µ³ is the -´%µ-module of all d matrices whose entries are
polynomials.
)
3 Any commutative ring with identity is a module over itself, that is, is
9
9
an -module. In this case, scalar multiplication is just multiplication by
9