Page 135 - Advanced Linear Algebra

Modules I: Basic Properties 119

(
Theorem 4.10 The second isomorphism theorem ) Let 4 be an -module
9
and let and be submodules of 4 . Then
:
;
:b ; :
š
; : q ;

Theorem 4.11 The third isomorphism theorem ) Let 4 be an -module and
(
9
suppose that :‹ ; are submodules of 4 . Then
4°: 4
š
;°: ;
Direct Sums and Direct Summands
The definition of direct sum of a family of submodules is a direct analog of the
definition for vector spaces.

Definition The external direct sum of -modules 4 ÁÃÁ4 , denoted by
9
Ä ^ 4 ^
4~ 4
is the -module whose elements are ordered -tuples

4 ~ ¸²# Áà Á# ³ “ #  4 Á ~ Áà Á ¹

with componentwise operations
²" ÁÃ Á" ³ b ²# ÁÃ Á# ³ ~ ²" b # ÁÃ Á" b # ³

and
²# Á Ã Á # ³ ~ ² # Á Ã Á # ³

for  9 .
We leave it to the reader to formulate the definition of external direct sums and
products for arbitrary families of modules, in direct analogy with the case of
vector spaces.
Definition An 9 -module 4 is the (internal ) direct sum of a family
< ~¸: “ 0¹ of submodules of 4, written

4~ or 4~ < :
0
if the following hold:
(
)
)
1 )(Join of the family 4 is the sum join of the family :
<
=~ :
0