What is "direct sum"

n. (context mathematics English) (l/en: coproduct) in some (l/en category categories), like (l/en abelian group abelian groups), (l/en topological space topological spaces) or (l/en module modules)

n. a union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets

The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum R ⊕ R, where R is the set of real numbers. R ⊕ R is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.

To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups A and B is another abelian group A ⊕ B consisting of the ordered pairs (a, b) where a ∈ A and b ∈ B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.

We can also form direct sums with any number of summands, for example A ⊕ B ⊕ C, provided A, B,  and C are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.

In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression xy) we use direct product.

In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. More generally, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are (A), the direct sum ⨁A is defined to be the set of tuples (a) with a ∈ A such that a = 0 for all but finitely many i. The direct sum ⨁A is contained in the direct product ∏A, but is usually strictly smaller when the index set I is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.