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, ${\displaystyle (x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})}$, which is the same as vector addition.

Given two structures ${\displaystyle A}$ and ${\displaystyle B}$, their direct sum is written as ${\displaystyle A\oplus B}$. Given an indexed family of structures ${\displaystyle A_{i}}$, indexed with ${\displaystyle i\in I}$, the direct sum may be written ${\textstyle A=\bigoplus _{i\in I}A_{i}}$. Each A_i 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 ${\displaystyle +}$ the phrase "direct sum" is used, while if the group operation is written ${\displaystyle *}$ 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.

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 ${\displaystyle \mathbb {R} }$ are defined, followed by ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$, the direct sum is said to be external.

If, on the other hand, some algebraic structure ${\displaystyle S}$ is defined, and ${\displaystyle S}$ is then defined as a direct sum of two substructures ${\displaystyle V}$ and ${\displaystyle W}$, the direct sum is said to be internal. In that case, each element of ${\displaystyle S}$ is expressible uniquely as an algebraic combination of an element of ${\displaystyle V}$ and an element of ${\displaystyle W}$. For an example of an internal direct sum, consider ${\displaystyle \mathbb {Z} _{6}}$ (the integers modulo six), whose elements are ${\displaystyle \{0,1,2,3,4,5\}}$. This is expressible as an internal direct sum ${\displaystyle \mathbb {Z} _{6}=\{0,2,4\}\oplus \{0,3\}}$.

The direct sum of abelian groups is a prototypical example of a direct sum. Given two such groups ${\displaystyle (A,\circ )}$ and ${\displaystyle (B,\bullet ),}$ their direct sum ${\displaystyle A\oplus B}$ is the same as their direct product. That is, the underlying set is the Cartesian product ${\displaystyle A\times B}$ and the group operation ${\displaystyle \,\cdot \,}$ is defined component-wise: $${\displaystyle \left(a_{1},b_{1}\right)\cdot \left(a_{2},b_{2}\right)=\left(a_{1}\circ a_{2},b_{1}\bullet b_{2}\right).}$$ This definition generalizes to direct sums of finitely many abelian groups.

For an arbitrary family of groups ${\displaystyle A_{i}}$ indexed by ${\displaystyle i\in I,}$ their direct sum^[2] $${\displaystyle \bigoplus _{i\in I}A_{i}}$$ is the subgroup of the direct product that consists of the elements ${\textstyle \left(a_{i}\right)_{i\in I}\in \prod _{i\in I}A_{i}}$ that have finite support, where, by definition, ${\displaystyle \left(a_{i}\right)_{i\in I}}$ is said to have finite support if ${\displaystyle a_{i}}$ is the identity element of ${\displaystyle A_{i}}$ for all but finitely many ${\displaystyle i.}$^[3] The direct sum of an infinite family ${\displaystyle \left(A_{i}\right)_{i\in I}}$ of non-trivial groups is a proper subgroup of the product group ${\textstyle \prod _{i\in I}A_{i}.}$

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.

An additive category is an abstraction of the properties of the category of modules.^[4][5] In such a category, finite products and coproducts agree, and the direct sum is either of them: cf. biproduct.

General case:^[2] 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.

However, the direct sum ${\displaystyle S_{3}\oplus \mathbb {Z} _{2}}$ (defined identically to the direct sum of abelian groups) is not a coproduct of the groups ${\displaystyle S_{3}}$ and ${\displaystyle \mathbb {Z} _{2}}$ in the category of groups. Therefore, for that category, a categorical direct sum is often called simply a coproduct to avoid any possible confusion.

The direct sum of group representations generalizes the direct sum of the underlying modules by adding a group action. Specifically, given a group ${\displaystyle G}$ and two representations ${\displaystyle V}$ and ${\displaystyle W}$ of ${\displaystyle G}$ (or, more generally, two ${\displaystyle G}$-modules), the direct sum of the representations is ${\displaystyle V\oplus W}$ with the action of ${\displaystyle g\in G}$ given component-wise, that is, $${\displaystyle g\cdot (v,w)=(g\cdot v,g\cdot w).}$$ Another equivalent way of defining the direct sum is as follows:

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

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

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

Some authors speak of the direct sum ${\displaystyle R\oplus S}$ of two rings when they mean the direct product ${\displaystyle R\times S}$, but that should be avoided^[6] since ${\displaystyle R\times S}$ does not receive natural ring homomorphisms from ${\displaystyle R}$ and ${\displaystyle S}$. In particular, the map ${\displaystyle R\to R\times S}$ sending ${\displaystyle r}$ to ${\displaystyle (r,0)}$ is not a ring homomorphism since it fails to send 1 to ${\displaystyle (1,1)}$ (assuming that ${\displaystyle 0\neq 1}$ in ${\displaystyle S}$). Thus, ${\displaystyle R\times S}$ 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.^[7] 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 ${\displaystyle (R_{i})_{i\in I}}$ 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.

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

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

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

If ${\displaystyle M}$ is a vector subspace of a real or complex vector space ${\displaystyle X}$, there is always another vector subspace ${\displaystyle N}$ of ${\displaystyle X,}$ called an algebraic complement of ${\displaystyle M}$ in ${\displaystyle X,}$ such that ${\displaystyle X}$ is the algebraic direct sum of ${\displaystyle M}$ and ${\displaystyle N}$, which happens if and only if the addition map ${\displaystyle M\times N\to X}$ 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 ${\displaystyle M}$ of ${\displaystyle X}$ is said to be a (topologically) complemented subspace of ${\displaystyle X}$ if there exists some vector subspace ${\displaystyle N}$ of ${\displaystyle X}$ such that ${\displaystyle X}$ is the topological direct sum of ${\displaystyle M}$ and ${\displaystyle N.}$ 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.

^{[clarification needed]}

The direct sum ${\textstyle \bigoplus _{i\in I}A_{i}}$ comes equipped with a projection homomorphism ${\textstyle \pi _{j}\colon \,\bigoplus _{i\in I}A_{i}\to A_{j}}$ for each j in I and a coprojection ${\textstyle \alpha _{j}\colon \,A_{j}\to \bigoplus _{i\in I}A_{i}}$ for each j in I.^[8] Given another algebraic structure ${\displaystyle B}$ (with the same additional structure) and homomorphisms ${\displaystyle g_{j}\colon A_{j}\to B}$ for every j in I, there is a unique homomorphism ${\textstyle g\colon \,\bigoplus _{i\in I}A_{i}\to B}$, called the sum of the g_j, such that ${\displaystyle g\alpha _{j}=g_{j}}$ for all j. Thus the direct sum is the coproduct in the appropriate category.

• Direct sum of groups
• Direct sum of permutations
• Direct sum of topological groups
• Restricted product
• Whitney sum
• Feferman–Vaught theorem