# Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum ${\displaystyle \mathbf {R} \oplus \mathbf {R} }$, where ${\displaystyle \mathbf {R} }$ is real coordinate space, is the Cartesian plane, ${\displaystyle \mathbf {R} ^{2}}$. To see how the 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 ${\displaystyle A}$ and ${\displaystyle B}$ is another abelian group ${\displaystyle A\oplus B}$ consisting of the ordered pairs ${\displaystyle (a,b)}$ where ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$. (Confusingly this ordered pair is also called the cartesian product of the two groups.) To add ordered pairs, we define the sum ${\displaystyle (a,b)+(c,d)}$ to be ${\displaystyle (a+c,b+d)}$; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of two vector spaces or two modules.

We can also form direct sums with any finite number of summands, for example ${\displaystyle A\oplus B\oplus C}$, provided ${\displaystyle A,B,}$ and ${\displaystyle C}$ are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is associative up to isomorphism. That is, ${\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}$ for any algebraic structures ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ of the same kind. The direct sum is also commutative up to isomorphism, i.e. ${\displaystyle A\oplus B\cong B\oplus A}$ for any algebraic structures ${\displaystyle A}$ and ${\displaystyle B}$ of the same kind.

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 ${\displaystyle xy}$) 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 ${\displaystyle (A_{i})_{i\in I}}$, the direct sum ${\displaystyle \bigoplus _{i\in I}A_{i}}$ is defined to be the set of tuples ${\displaystyle (a_{i})_{i\in I}}$ with ${\displaystyle a_{i}\in A_{i}}$ such that ${\displaystyle a_{i}=0}$ for all but finitely many i. The direct sum ${\displaystyle \bigoplus _{i\in I}A_{i}}$ is contained in the direct product ${\displaystyle \prod _{i\in I}A_{i}}$, but is usually strictly smaller when the index set ${\displaystyle I}$ is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.[1]

## Examples

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 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 ${\displaystyle \textstyle A=\bigoplus _{i\in I}A_{i}}$. 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 ${\displaystyle +}$ 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.

### Internal and external direct sums

A distinction is made between internal and external direct sums, though the two are isomorphic. If the factors are defined first, and then the direct sum is defined in terms of the factors, 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 ${\displaystyle \mathbb {Z} _{6}=\{0,2,4\}\oplus \{0,3\}}$.

## Types of 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 ${\displaystyle (A,\circ )}$ 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 infinite family of abelian groups ${\displaystyle A_{i}}$ for ${\displaystyle i\in I,}$ the direct sum

${\displaystyle \bigoplus _{i\in I}A_{i}}$
is a proper subgroup of the direct product. It consists of the elements ${\displaystyle \textstyle (a_{i})\in \prod _{j\in I}A_{j}}$ such that ${\displaystyle a_{i}}$ is the identity element of ${\displaystyle A_{i}}$ for all but finitely many ${\displaystyle i.}$[2]

### 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 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[3] 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.[4] 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 in categories

An additive category is an abstraction of the properties of the category of modules.[5][6] In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case:[7] 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.

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.[8] So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.

### Direct sum of matrices

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

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

## Homomorphisms

The direct sum ${\displaystyle \bigoplus _{i\in I}A_{i}}$ comes equipped with a projection homomorphism ${\displaystyle \pi _{j}\colon \,\bigoplus _{i\in I}A_{i}\to A_{j}}$ for each j in I and a coprojection ${\displaystyle \alpha _{j}\colon \,A_{j}\to \bigoplus _{i\in I}A_{i}}$ for each j in I.[9] 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 ${\displaystyle g\colon \,\bigoplus _{i\in I}A_{i}\to B}$, called the sum of the gj, such that ${\displaystyle g\alpha _{j}=g_{j}}$ for all j. Thus the direct sum is the coproduct in the appropriate category.

## Notes

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. ^ Math StackExchange on direct sum of rings vs. direct product of rings.
4. ^ Lang 2002, section I.11
5. ^ "p.45"
6. ^ "Appendix" (PDF). Archived from the original (PDF) on 2006-09-17. Retrieved 2014-01-14.
7. ^
8. ^ "Counterexamples for products and coproduct". Planetmath. Retrieved 2021-07-23.
9. ^ Heunen, Chris (2009). Categorical Quantum Models and Logics. Pallas Proefschriften. Amsterdam University Press. p. 26. ISBN 978-9085550242.