The module direct sum of modules and is the module

where all algebraic operations are defined componentwise. In particular, suppose that and are left -modules, then

and

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.

See also

Coproduct, Direct Sum, Group Direct Product, Module

Portions of this entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Module Direct Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModuleDirectSum.html

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