Editing Direct sum of modules (section)

{{Short description|Operation in abstract algebra}}
{{For|the broader use of the term in mathematics|Direct sum}}
 
In [[abstract algebra]], the '''direct sum''' is a construction which combines several [[module (mathematics)|modules]] into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a [[coproduct]]. Contrast with the [[direct product]], which is the [[dual (category theory)|dual]] notion.

The most familiar examples of this construction occur when considering [[vector space]]s (modules over a [[field (mathematics)|field]]) and [[abelian group]]s (modules over the ring '''Z''' of [[integer]]s). The construction may also be extended to cover [[Banach space]]s and [[Hilbert space]]s.

See the article [[decomposition of a module]] for a way to write a module as a direct sum of submodules.