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 Ai, indexed with iI 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:

    The most passionate, consistent, extreme and implacable enemy of the Enlightenment and ... all forms of rationalism ... was Johann Georg Hamann. His influence, direct and indirect, upon the romantic revolt against universalism and scientific method ... was considerable and perhaps crucial.
    Isaiah Berlin (b. 1909)

    Lest darkness fall and time fall
    In a long night when learned arteries
    Mounting the ice and sum of barbarous time
    Shall yield, without essence, perfect accident.
    We are the eyelids of defeated caves.
    Allen Tate (1899–1979)