Internal and external direct sums

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

If, on the other hand, we first define some algebraic structure ${\displaystyle S}$ and then write ${\displaystyle S}$ as a direct sum of two substructures ${\displaystyle V}$ and ${\displaystyle W}$, then the direct sum is said to be internal. In this 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 ${\displaystyle \mathbb {Z} _{6}=\{0,2,4\}\oplus \{0,3\}}$.

Direct sum of abelian groups

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 ${\textstyle \prod _{i\in I}A_{i}.}$

Direct sum of modules

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

The most familiar examples of this construction occur when 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

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, direct sum is a coproduct. This is also true in the category of modules.

Direct sum of 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 ${\displaystyle G}$ 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 we treat ${\displaystyle V}$ and ${\displaystyle W}$ as modules over the group ring ${\displaystyle kG}$, where ${\displaystyle k}$ is the field, then the direct sum of the representations ${\displaystyle V}$ and ${\displaystyle W}$ is equal to their direct sum as ${\displaystyle kG}$ modules.

Direct sum of rings

Some authors will speak of the direct sum ${\displaystyle R\oplus S}$ of two rings when they mean the direct product ${\displaystyle R\times S}$, but this 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.)

Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If ${\displaystyle (R_{i})_{i\in I}}$ is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, that is, a ring without a multiplicative identity.

Direct sum of matrices

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).

$${\displaystyle \mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}\mathbf {A} &0\\0&\mathbf {B} \end{bmatrix}}.}$$

Direct sum of topological vector spaces

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 ${\displaystyle X.}$ This 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 ${\displaystyle X}$ then there always exists 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.