Editing Direct sum of modules (section)

=== Construction for two abelian groups ===

For [[abelian group]]s ''G'' and ''H'' which are written additively, the [[direct product]] of ''G'' and ''H'' is also called a direct sum {{harv|Mac Lane|Birkhoff|1999|loc=§V.6}}. Thus the [[Cartesian product]] ''G'' × ''H'' is equipped with the structure of an abelian group by defining the operations componentwise:
: (''g''<sub>1</sub>, ''h''<sub>1</sub>) + (''g''<sub>2</sub>, ''h''<sub>2</sub>) = (''g''<sub>1</sub> + ''g''<sub>2</sub>, ''h''<sub>1</sub> + ''h''<sub>2</sub>)

for ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'', and ''h''<sub>1</sub>, ''h''<sub>2</sub> in ''H''.

Integral multiples are similarly defined componentwise by
: ''n''(''g'', ''h'') = (''ng'', ''nh'')

for ''g'' in ''G'', ''h'' in ''H'', and ''n'' an [[integer]]. This parallels the extension of the scalar product of vector spaces to the direct sum above.

The resulting abelian group is called the ''direct sum'' of ''G'' and ''H'' and is usually denoted by a plus symbol inside a circle:
<math display=block>G \oplus H</math>

It is customary to write the elements of an ordered sum not as ordered pairs (''g'', ''h''), but as a sum ''g'' + ''h''.

The [[subgroup]] ''G'' × {0} of ''G'' ⊕ ''H'' is isomorphic to ''G'' and is often identified with ''G''; similarly for {0} × ''H'' and ''H''. (See [[Direct_sum_of_modules#Internal_direct_sum|''internal direct sum'']] below.) With this identification, it is true that every element of ''G'' ⊕ ''H'' can be written in one and only one way as the sum of an element of ''G'' and an element of ''H''. The [[rank of an abelian group|rank]] of ''G'' ⊕ ''H'' is equal to the sum of the ranks of ''G'' and ''H''.

This construction readily generalises to any [[finite set|finite]] number of abelian groups.