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*is called a

_{i}**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**.

Read more about Direct Sum: Direct Sum of Abelian Groups, Direct Sum of Rings, Internal Direct Sum

### Famous quotes containing the words direct and/or sum:

“No amount of charters, *direct* primaries, or short ballots will make a democracy out of an illiterate people.”

—Walter Lippmann (1889–1974)

“but Overall is beyond me: is the *sum* of these events

I cannot draw, the ledger I cannot keep, the accounting

beyond the account:”

—Archie Randolph Ammons (b. 1926)