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 Ai, indexed with i ∈ I from an index set I, one may write their direct sum as . Each Ai 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.
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:
“You see how this House of Commons has begun to verify all the ill prophecies that were made of it—low, vulgar, meddling with everything, assuming universal competency, and flattering every base passion—and sneering at everything noble refined and truly national. The direct tyranny will come on by and by, after it shall have gratified the multitude with the spoil and ruin of the old institutions of the land.”
—Samuel Taylor Coleridge (1772–1834)
“The real risks for any artist are taken ... in pushing the work to the limits of what is possible, in the attempt to increase the sum of what it is possible to think. Books become good when they go to this edge and risk falling over it—when they endanger the artist by reason of what he has, or has not, artistically dared.”
—Salman Rushdie (b. 1947)