The direct sum of modules 
and 
is the module
 |
(1)
|
where all algebraic operations are defined componentwise. In particular, suppose that 
and 
are left 
-modules,
then
 |
(2)
|
and
 |
(3)
|
where 
is an element of the ring 
. The direct sum of an arbitrary family of modules
over the same ring is also defined. If 
is the indexing set for the family of modules,
then the direct sum is represented by the collection of functions with finite support
from 
to the union of all these modules such that the function
sends 
to an element in the module indexed by 
.
The dimension of a direct sum is the sum of the dimensions of the quantities summed. The significant property of the direct sum is that it is the coproduct
in the category of modules.
This general definition gives as a consequence the definition of the direct sum 
of Abelian
groups 
and 
(since they are 
-modules,
i.e., modules over the integers)
and the direct sum of vector spaces (since they are
modules over a field). Note
that the direct sum of Abelian groups is the same as the group
direct product, but that the term direct sum is not used for groups which are
non-Abelian.
Whenever 
is a module, with module
homomorphisms 
and 
,
then there is a module homomorphism 
, given by 
. Note that this map is well-defined
because addition in modules is commutative. Sometimes direct sum is preferred over
direct product when the coproduct property is emphasized.
