Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-III
63
n.v =
R
i.e. abelian group ≡ Z -module
2. A vector space V over a field F is an F-module
3. A linear Vector space
is an M n
(F)-module if A.V is usual product, where
FI
x1
A = ( ai ) n n ∈ Mn
F
j × ( ) , v x
=
2
Μ
G
, the column vector v of length n from
x J
F
n .
H2
K
n×
1
4. Let V be a vector space over the field F. T : V –––→ V is a linear operator. V can be made F[ x] -
module by defining
f ( x). v = f ( T) v,
Free modules
.
( vm SF RM f1 v∈
( + + ...... + v, n + integer
n
= ∈R ≠ xV ) r( φ0∈
()
) F)
[ x],
1v1 + ........ Let + r k
vM k isa be a ve module over a ring R, and S be a subset of M.S is said to be a basis of M if
→ n times ↵
1.
S|
( − v) + ( − v) + + ( − v) = − ( m.v),
when n = −m
2. S generates M
= ( − m).
v and m is + ve integer
3. S is linearly independent.
T|
O if n = zero
If S is a basis of M, then in particular , if and every element of M has a unique expression
as a linear combination of elements of S.
If R is a ring, then as a module over itself, R admits a basis, consisting of unit element 1
Free Module
A module which admits a basis. We include in definition, the zero module also for free module
Remarks
1. An ordered set ( m 1 , m 2 )....., m k ) of element of a module M is said to generate (or span) if every
m ∈ M is a linear combination :
, r i ∈R
Here elements v i
are called generators. A module M is said to be finitely generated if there exists
a finite set of generators.
A Z-module M is finitely generated ⇔ it is finitely generated abelian group.
2. Consider,
.