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:
“Irony, forsooth! Guard yourself, Engineer, from the sort of irony that thrives up here; guard yourself altogether from taking on their mental attitude! Where irony is not a direct and classic device of oratory, not for a moment equivocal to a healthy mind, it makes for depravity, it becomes a drawback to civilization, an unclean traffic with the forces of reaction, vice and materialism.”
—Thomas Mann (1875–1955)
“The history of literature—take the net result of Tiraboshi, Warton, or Schlegel,—is a sum of a very few ideas, and of very few original tales,—all the rest being variation of these.”
—Ralph Waldo Emerson (1803–1882)