Page 125 - Advanced Linear Algebra

Chapter 4

Modules I: Basic Properties

Motivation

Let = be a vector space over a field - and let  B ² = ³ . Then for any

polynomial ²%³  -´%µ , the operator ² ³ is well-defined. For instance, if

²%³ ~ b % b % , then
² ³ ~ b b

where is the identity operator and is the threefold composition kk .
Thus, using the operator we can define the product of a polynomial

²%³ -´%µ and a vector # = by
²%³# ~ ² ³²#³ (4.1 )

This product satisfies the usual properties of scalar multiplication, namely, for
all ²%³Á ²%³  -´%µ and "Á #  = ,
²%³²" b #³ ~ ²%³" b ²%³#
² ²%³ b ²%³³" ~ ²%³" b ²%³"
´ ²%³ ²%³µ" ~ ²%³´ ²%³"µ
" ~ "

Thus, for a fixed B ²= ³ , we can think of = as being endowed with the
operations of addition and multiplication of an element of by a polynomial in
=
-´%µ. However, since -´%µ is not a field, these two operations do not make =
into a vector space. Nevertheless, the situation in which the scalars form a ring
but not a field is extremely important, not only in this context but in many
others.
Modules

Definition Let 9 be a commutative ring with identity, whose elements are
called scalars . An -module ( ) is a nonempty set 4 ,
9
or a module over 9