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.
Portions of this entry contributed by Todd Rowland
