Direct sum

The symbol $\oplus \!$ denotes direct sum; it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction.

In mathematics, given a pair of mathematical objects of some given kind, one can often define a direct sum of the objects, creating a new object of the same kind. The direct sum of a pair of objects is generally the Cartesian product (or some subset of it) of the pair of underlying sets corresponding to the objects, together with a suitably defined mathematical structure induced by the structure of the objects. Informally, the direct sum of a pair of objects is the "smallest" combination of them that does not introduce any extra constraints on the structure.

For example, the xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y spaces only intersect at the zero vector, i.e., the origin. In terms of coordinates, the direct sum combines the x axis coordinate (x) and y axis coordinate (y) into an xy coordinate (x,y). This representation makes it clear that the direct sum is a Cartesian product of the two sets corresponding to the x and y axes. In this example, the mathematical structure induced is just vector addition: $(x_1,y_1) + (x_2,y_2) = (x_1+x_2, y_1 + y_2).$

Other examples of direct sums 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. More abstractly, the direct sum is often, but not always, the coproduct in the category of mathematical objects in question.

Given two objects A and B, their direct sum is written as $A\oplus B$. Given an indexed family of objects A_i, indexed with i ∈ I from an index set I, one may write their direct sum as $\textstyle A=\bigoplus_{i\in I}A_i$. Each A_i is called a direct summand of A.

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, as in the xy-plane example above, the direct sum can be referred to as an internal direct sum.

Direct sum of abelian groups

The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups (A, ∗) and (B, ·), their direct sum A ⊕ B is the same as their direct product, i.e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise:

(a₁, b₁) ○ (a₂, b₂) = (a₁ ∗ a₂, b₁ · b₂).

This definition generalizes to direct sums of finitely many abelian groups.

For an infinite family of abelian groups A_i for i ∈ I, the direct sum

$\bigoplus_{i\in I}A_i$

is a proper subgroup of the direct product. It consists of the elements $\textstyle (a_i)\in\prod_{j\in I}A_j$ such that a_i is the identity element of A_i for all but finitely many i.

In this case, the direct sum is indeed the coproduct in the category of abelian groups.

Direct sum of modules (e.g. vector spaces)

Main article: Direct sum of modules

Direct sum of representations

Group representations

The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it. Specifically, given a group G and two representations V and W of G (or, more generally, two G-modules), the direct sum of the representations is V ⊕ W with the action of g ∈ G given component-wise, i.e.

g·(v, w) = (g·v, g·w).

Direct sum of rings

Main article: Product of rings

Given a finite family of rings R₁, ..., R_n, the direct product of the R_i is sometimes called the direct sum.

Note that in the category of commutative rings, the direct sum is not the coproduct. Instead, the coproduct is the tensor product of rings.

Internal direct sum

An internal direct sum is essentially a direct sum of subobjects of an object.

For example, the real vector space R² = {(x, y) : x, y ∈ R} is the internal direct sum of the x-axis {(x, 0) : x ∈ R} and the y-axis {(0, y) : y ∈ R}, and the sum of (x, 0) and (0, y) is the "internal" sum in the vector space R²; thus, this is an internal direct sum. More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if

1. U + W = {u + w : u ∈ U, w ∈ W} = V, and
2. if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0.

In other words, every element of V can be written uniquely as the sum of an element in U with an element of W

Another case is that of abelian groups. For example, the Klein four-group V = {e, a, b, ab} is the (internal) direct sum of the cyclic subgroups <a> and <b>.

By contrast, a direct sum of two objects which are not subobjects of a common object is an external direct sum. Note however that "external direct sum" is also used to refer to an infinite direct sum of groups, to contrast with the (larger) direct product.

See also

• Restricted product

References

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR1878556