Direct Sum

In mathematics, given a pair of objects of some given kind, one can often define a direct sum of them, giving a new object of the same kind. This is generally the Cartesian product of the underlying sets (or some subset of it), together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question.

Given two objects A and B, their direct sum is written as . Given a indexed family of objects A_i, indexed with i ∈ I from an index set I, one may write their direct sum as . Each A_i is called a direct summand of A.

Examples include the direct sum of abelian groups, the direct sum of modules, the direct sum of rings, the direct sum of matrices, and the direct sum of topological spaces.

A related concept is that of the direct product, which is sometimes the same as the direct sum, but at other times can be entirely different.

In cases where an object is expressed as a direct sum of subobjects, the direct sum can be referred to as an internal direct sum.