Direct sum

Examples

The xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces: the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise; that is, (x_1,y_1) + (x_2,y_2) = (x_1+x_2, y_1 + y_2), which is the same as vector addition.

Given two structures A and B, their direct sum is written as A\oplus B. Given an indexed family of structures A_i, indexed with i \in I, the direct sum may be written A=\bigoplus_{i\in I}A_i. Each Ai is called a direct summand of A. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as + the phrase "direct sum" is used, while if the group operation is written * the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.

Internal and external direct sums

A distinction is made between internal and external direct sums though both are isomorphic. If the summands are defined first, and the direct sum is then defined in terms of the summands, there is an external direct sum. For example, if the real numbers \mathbb{R} are defined, followed by \mathbb{R} \oplus \mathbb{R}, the direct sum is said to be external.

If, on the other hand, some algebraic structure S is defined, and S is then defined as a direct sum of two substructures V and W, the direct sum is said to be internal. In that case, each element of S is expressible uniquely as an algebraic combination of an element of V and an element of W. For an example of an internal direct sum, consider \mathbb Z_6 (the integers modulo six), whose elements are {0, 1, 2, 3, 4, 5}. This is expressible as an internal direct sum \mathbb Z_6 = {0, 2, 4} \oplus {0, 3}.

Types of direct sums

Direct sum of abelian groups

Main article: Direct product of groups

The direct sum of abelian groups is a prototypical example of a direct sum. Given two such groups (A, \circ) and (B, \bullet), their direct sum A \oplus B is the same as their direct product. That is, the underlying set is the Cartesian product A \times B and the group operation ,\cdot, is defined component-wise: \left(a_1, b_1\right) \cdot \left(a_2, b_2\right) = \left(a_1 \circ a_2, b_1 \bullet b_2\right). This definition generalizes to direct sums of finitely many abelian groups.

For an arbitrary family of groups A_i indexed by i \in I, their direct sum \bigoplus_{i \in I} A_i is the subgroup of the direct product that consists of the elements \left(a_i\right){i \in I} \in \prod{i \in I} A_i that have finite support, where, by definition, \left(a_i\right){i \in I} is said to have finite support if a_i is the identity element of A_i for all but finitely many i. The direct sum of an infinite family \left(A_i\right){i \in I} of non-trivial groups is a proper subgroup of the product group \prod_{i \in I} A_i.

Direct sum of modules

Main article: Direct sum of modules

The direct sum of modules is a construction that combines several modules into a new module.

The most familiar examples of that construction occur in considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.

Direct sum in categories

Main article: Coproduct

An additive category is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree, and the direct sum is either of them: cf. biproduct.

More generally, in category theory the direct sum is often but not always the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, the direct sum is a coproduct. That is also true in the category of modules.

In contrast, in the category of (possibly-nonabelian) groups, a "direct sum" can be defined identically to the direct sum of abelian groups, but it does not form a coproduct in that category. For instance, S_3 \oplus \Z_2 is not a coproduct of the groups S_3 and \Z_2.

Direct sum of group representations

The direct sum of group representations generalizes the direct sum of the underlying modules by adding a group action. 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 \oplus W with the action of g \in G given component-wise, that is, g \cdot (v, w) = (g \cdot v, g \cdot w). Another equivalent way of defining the direct sum is as follows:

Given two representations (V, \rho_V) and (W, \rho_W) the vector space of the direct sum is V \oplus W and the homomorphism \rho_{V \oplus W} is given by \alpha \circ (\rho_V \times \rho_W), where \alpha: GL(V) \times GL(W) \to GL(V \oplus W) is the natural map obtained by coordinate-wise action as above.

Furthermore, if V,,W are finite dimensional, then, given a basis of V,,W, \rho_V and \rho_W are matrix-valued. In this case, \rho_{V \oplus W} is given as g \mapsto \begin{pmatrix}\rho_V(g) & 0 \ 0 & \rho_W(g)\end{pmatrix}.

Moreover, if V and W are treated as modules over the group ring kG, where k is the field, the direct sum of the representations V and W is equal to their direct sum as kG modules.

Direct sum of rings

Main article: Product of rings

Some authors speak of the direct sum R \oplus S of two rings when they mean the direct product R \times S, but that should be avoided since R \times S does not receive natural ring homomorphisms from R and S. In particular, the map R \to R \times S sending r to (r, 0) is not a ring homomorphism since it fails to send 1 to (1, 1) (assuming that 0 \neq 1 in S). Thus, R \times S is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings. In the category of rings, the coproduct is given by a construction similar to the free product of groups.)

The use of direct sum terminology and notation is especially problematic in dealing with infinite families of rings. If (R_i)_{i \in I} is an infinite collection of nontrivial rings, the direct sum of the underlying additive groups may be equipped with termwise multiplication, but that produces a rng, a ring without a multiplicative identity.

Direct sum of matrices

For any arbitrary matrices \mathbf{A} and \mathbf{B}, the direct sum \mathbf{A} \oplus \mathbf{B} is defined as the block diagonal matrix of \mathbf{A} and \mathbf{B} if both are square matrices (and to an analogous block matrix, if not). \mathbf{A} \oplus \mathbf{B} = \begin{bmatrix} \mathbf{A} & 0 \ 0 & \mathbf{B} \end{bmatrix}.

Alternatively, the forms \left[\begin{matrix}\mathbf{A} \ \mathbf{B}\end{matrix}\right] or \left[\begin{matrix} \mathbf{A} & \mathbf{B}\end{matrix}\right] may also be encountered in the literature and are isomorphic to the aforementioned block form.

Direct sum of topological vector spaces

Main article: Complemented subspace, Direct sum of topological groups

A topological vector space (TVS) X, such as a Banach space, is said to be a topological direct sum of two vector subspaces M and N if the addition map \begin{alignat}{4} \ ;&& M \times N &&;\to ;& X \[0.3ex] && (m, n) &&;\mapsto;& m + n \ \end{alignat} is an isomorphism of topological vector spaces (meaning that this linear map is a bijective homeomorphism) in which case M and N are said to be topological complements in X. That is true if and only if when considered as additive topological groups (so scalar multiplication is ignored), X is the topological direct sum of the topological subgroups M and N. If this is the case and if X is Hausdorff then M and N are necessarily closed subspaces of X.

If M is a vector subspace of a real or complex vector space X, there is always another vector subspace N of X, called an algebraic complement of M in X, such that X is the algebraic direct sum of M and N, which happens if and only if the addition map M \times N \to X is a vector space isomorphism.

In contrast to algebraic direct sums, the existence of such a complement is no longer guaranteed for topological direct sums.

A vector subspace M of X is said to be a (topologically) complemented subspace of X if there exists some vector subspace N of X such that X is the topological direct sum of M and N. A vector subspace is called uncomplemented if it is not a complemented subspace. For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented. Every closed vector subspace of a Hilbert space is complemented. But every Banach space that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.

Homomorphisms

The direct sum \bigoplus_{i \in I} A_i comes equipped with a projection homomorphism \pi_j \colon , \bigoplus_{i \in I} A_i \to A_j for each j in I and a coprojection \alpha_j \colon , A_j \to \bigoplus_{i \in I} A_i for each j in I. Given another algebraic structure B (with the same additional structure) and homomorphisms g_j \colon A_j \to B for every j in I, there is a unique homomorphism g \colon , \bigoplus_{i \in I} A_i \to B, called the sum of the g**j, such that g \alpha_j =g_j for all j. Thus the direct sum is the coproduct in the appropriate category.

Notes

References

References

1. [[Thomas W. Hungerford]], ''Algebra'', p.60, Springer, 1974, {{ISBN. 0387905189
2. Joseph J. Rotman, ''The Theory of Groups: an Introduction'', p. 177, Allyn and Bacon, 1965
3. ""p.45"".
4. "Appendix".
5. "Direct Sum".
6. [https://math.stackexchange.com/q/345501 Math StackExchange] on direct sum of rings vs. direct product of rings.{{User-generated source. (November 2025)
7. {{harvnb. Lang. 2002, section I.11
8. Heunen, Chris. (2009). "Categorical Quantum Models and Logics". Amsterdam University Press.