The module 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 index 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 
.
For vector spaces, the dimension of a direct sum is the sum of the dimensions
of the vector spaces 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).
For a finite family of Abelian
groups, the direct sum is the same as the group
direct product. For an infinite family,
the direct sum consists of the elements of the group
direct product with finite support.
The term direct sum is most often used for Abelian groups,
but is sometimes also used for non-Abelian groups.
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.
