Direct sum

Operation in abstract algebra composing objects into "more complicated" objects

The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.

The direct sum of modules is a construction which combines several modules into a new module.

The most familiar examples of this construction occur when considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.

An additive category is an abstraction of the properties of the category of modules.[4][5] In such a category, finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case:[2] In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.