Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-III
75
Example 5
The direct sum of free modules over R is a force module over R, its basis being the union of the bases of the
direct summands.
Example 6
A submodule of a free module over a ring R, is not necessarily a free module. However, every submodule of
a free module over a principal ideal domain (P.I.D.) is free.
We mention the following results without proof (can be seen in a standard book of algebra):–
Results
1. Let M be a free module over a P.I.D. with a finite basis
M is free and has a basis of < n elements.
. Then every submodule N &
2. From (1) we can deduce that a submodule N & a finitely generated Module M over a P.I.D. is finitely
generated.
Recall that for each finite abelian group G ≠ ( 0 ) there is exactly one list
m i > 1, each a multiple of the next, for which there is an isomorphism.
G ≅ Z ⊕............
⊕Z
m1
m k
the first integer m 1
is the least +ve integer m = m 1
with mG = ( 0 ) and the product
of integers
Example 7
mZ x 1 , m,........
x, 2
Z ,.......... ⊕m k Zm x, ZThe n
= ⊕( possible GZ ) , Z ⊕abelian Z group of order 36 are
36 1 2 18 2k
120
3 6 6
No two of these group are isomorphic.