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Thedirect sum is anoperation betweenstructures inabstract algebra, a branch ofmathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of twoabelian 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${\displaystyle b\in B}$. To addordered pairs, the sum is defined${\displaystyle (a,b)+(c,d)}$ to be${\displaystyle (a+c,b+d)}$; in other words, addition is defined coordinate-wise. For example, the direct sum${\displaystyle \mathbb{R}\oplus \mathbb{R}}$, where${\displaystyle \mathbb{R}}$ isreal coordinate space, is theCartesian plane,${\displaystyle {\mathbb{R}}^2}$. A similar process can be used to form the direct sum of twovector spaces or twomodules.

Direct sums can also be formed 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). That relies on the fact that the direct sum isassociativeup toisomorphism. 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 alsocommutative 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.

The direct sum of finitely many abelian groups, vector spaces, or modules is canonicallyisomorphic to the correspondingdirect product. That is false, however, for some algebraic objects like nonabelian groups.

In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic even for abelian groups, vector spaces, or modules. For example, consider the direct sum and the direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is 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. Often, 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$${\displaystyle \underset{i\in I}{⨁}{A}_i}$$ 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 manyi. The direct sum$\underset{i\in I}{⨁}{A}_i$ is contained in thedirect product$\prod _{i\in I}{A}_i$, but is strictly smaller when theindex set${\displaystyle I}$ is infinite, because an element of the direct product can have infinitely many nonzero coordinates.^[1]

Thexy-plane, a two-dimensionalvector space, can be thought of as the direct sum of two one-dimensional vector spaces: thex andy axes. In this direct sum, thex andy 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 anindexed family of structures${\displaystyle A_i}$, indexed with${\displaystyle i\in I}$, the direct sum may be written$A=\underset{i\in I}{⨁}{A}_i$. EachA_i is called adirect summand ofA. 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${\displaystyle \ast }$ 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${\displaystyle {\mathbb{Z}}_6=\{0,2,4\}\oplus \{0,3\}}$.

Thedirect sum ofabelian groups is a prototypical example of a direct sum. Given two suchgroups${\displaystyle (A,\circ )}$ and${\displaystyle (B,\bullet ),}$ their direct sum${\displaystyle A\oplus B}$ is the same as theirdirect product. That is, the underlying set is theCartesian 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${\displaystyle i\in I,}$ theirdirect sum^[2]$${\displaystyle \underset{i\in I}{⨁}{A}_i}$$is thesubgroup 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 finitesupport, where, by definition,${\displaystyle {\left(a_i\right)}_{i\in I}}$ is said to havefinite 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 aproper subgroup of the product group$\prod _{i\in I}{A}_i.$

Thedirect sum of modules is a construction that combines severalmodules into a new module.

The most familiar examples of that construction occur in consideringvector spaces, which are modules over afield. The construction may also be extended toBanach spaces andHilbert spaces.

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

General case:^[2]Incategory theory thedirect sum is often but not always the coproduct in thecategory 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 thecategory of groups. Therefore, for that category, a categorical direct sum is often called simply a coproduct to avoid any possible confusion.

Thedirect sum of group representations generalizes thedirect sum of the underlying modules by adding agroup action. Specifically, given agroup${\displaystyle G}$ and tworepresentations${\displaystyle V}$ 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

Moreover, if${\displaystyle V}$ and${\displaystyle W}$ are treated as modules over thegroup 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 thedirect 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${\displaystyle 0\ne 1}$ in${\displaystyle S}$). Thus,${\displaystyle R\times S}$ is not a coproduct in thecategory of rings, and should not be written as a direct sum. (The coproduct in thecategory of commutative rings is thetensor product of rings.^[7] In the category of rings, the coproduct is given by a construction similar to thefree 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 arng, 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 theblock diagonal matrix of${\displaystyle \mathbf{A}}$ and${\displaystyle \mathbf{B}}$ if both are square matrices (and to an analogousblock matrix, if not).

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

Atopological vector space (TVS)${\displaystyle X,}$ such as aBanach space, is said to be atopological direct sum of two vector subspaces${\displaystyle M}$ and${\displaystyle N}$ if the addition mapis anisomorphism of topological vector spaces (meaning that thislinear map is abijectivehomeomorphism) in which case${\displaystyle M}$ and${\displaystyle N}$ are said to betopological complements in${\displaystyle X.}$ That is true if and only if when considered asadditivetopological groups (so scalar multiplication is ignored),${\displaystyle X}$ is thetopological direct sum of the topological subgroups${\displaystyle M}$ and${\displaystyle N.}$ If this is the case and if${\displaystyle X}$ isHausdorff then${\displaystyle M}$ and${\displaystyle N}$ are necessarilyclosed 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 analgebraic complement of${\displaystyle M}$ in${\displaystyle X,}$ such that${\displaystyle X}$ is thealgebraic direct sum of${\displaystyle M}$ and${\displaystyle N}$, which happens if and only if the addition map${\displaystyle M\times N\to X}$ is avector 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 calleduncomplemented 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 aHilbert space is complemented. But everyBanach space that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.

^{[clarification needed]}

The direct sum$\underset{i\in I}{⨁}{A}_i$ comes equipped with aprojectionhomomorphism${\pi }_j:\underset{i\in I}{⨁}{A}_i\to A_j$ for eachj inI and acoprojection${\alpha }_j:A_j\to \underset{i\in I}{⨁}{A}_i$ for eachj inI.^[8] Given another algebraic structure${\displaystyle B}$ (with the same additional structure) and homomorphisms${\displaystyle g_j:A_j\to B}$ for everyj inI, there is a unique homomorphism$g:\underset{i\in I}{⨁}{A}_i\to B$, called the sum of theg_j, such that${\displaystyle g{\alpha }_j=g_j}$ for allj. Thus the direct sum is thecoproduct in the appropriatecategory.

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

• Lang, Serge (2002),Algebra,Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556,Zbl 0984.00001

• Abstract algebra

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