Inabstract algebra, thedirect sum is a construction which combines severalmodules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of acoproduct. Contrast with thedirect product, which is thedual notion.

The most familiar examples of this construction occur when consideringvector spaces (modules over afield) andabelian groups (modules over the ringZ ofintegers). The construction may also be extended to coverBanach spaces andHilbert spaces.

See the articledecomposition of a module for a way to write a module as a direct sum of submodules.

We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth.

SupposeV andW arevector spaces over thefieldK. TheCartesian productV ×W can be given the structure of a vector space overK (Halmos 1974, §18) by defining the operations componentwise:

• (v₁,w₁) + (v₂,w₂) = (v₁ +v₂,w₁ +w₂)
• α (v,w) = (αv,αw)

forv,v₁,v₂ ∈V,w,w₁,w₂ ∈W, andα ∈K.

The resulting vector space is called thedirect sum ofV andW and is usually denoted by a plus symbol inside a circle:$${\displaystyle V\oplus W}$$

It is customary to write the elements of an ordered sum not as ordered pairs (v,w), but as a sumv +w.

The subspaceV × {0} ofV ⊕W is isomorphic toV and is often identified withV; similarly for {0} ×W andW. (Seeinternal direct sum below.) With this identification, every element ofV ⊕W can be written in one and only one way as the sum of an element ofV and an element ofW. Thedimension ofV ⊕W is equal to the sum of the dimensions ofV andW. One elementary use is the reconstruction of a finite vector space from any subspaceW and its orthogonal complement:$${\displaystyle {\mathbb{R}}^n=W\oplus W^{\perp }}$$

This construction readily generalizes to anyfinite number of vector spaces.

Forabelian groupsG andH which are written additively, thedirect product ofG andH is also called a direct sum (Mac Lane & Birkhoff 1999, §V.6). Thus theCartesian productG ×H is equipped with the structure of an abelian group by defining the operations componentwise:

(g₁,h₁) + (g₂,h₂) = (g₁ +g₂,h₁ +h₂)

forg₁,g₂ inG, andh₁,h₂ inH.

Integral multiples are similarly defined componentwise by

n(g,h) = (ng,nh)

forg inG,h inH, andn aninteger. This parallels the extension of the scalar product of vector spaces to the direct sum above.

The resulting abelian group is called thedirect sum ofG andH and is usually denoted by a plus symbol inside a circle:$${\displaystyle G\oplus H}$$

It is customary to write the elements of an ordered sum not as ordered pairs (g,h), but as a sumg +h.

ThesubgroupG × {0} ofG ⊕H is isomorphic toG and is often identified withG; similarly for {0} ×H andH. (Seeinternal direct sum below.) With this identification, it is true that every element ofG ⊕H can be written in one and only one way as the sum of an element ofG and an element ofH. Therank ofG ⊕H is equal to the sum of the ranks ofG andH.

This construction readily generalises to anyfinite number of abelian groups.

One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of twomodules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows (Bourbaki 1989, §II.1.6).

LetR be a ring, and {M_i : i ∈ I} afamily of leftR-modules indexed by thesetI. Thedirect sum of {M_i} is then defined to be the set of all sequences${\displaystyle ({\alpha }_i)}$ where${\displaystyle {\alpha }_i\in M_i}$ and${\displaystyle {\alpha }_i=0}$ forcofinitely many indicesi. (Thedirect product is analogous but the indices do not need to cofinitely vanish.)

It can also be defined asfunctions α fromI to thedisjoint union of the modulesM_i such that α(i) ∈ M_i for alli ∈I and α(i) = 0 forcofinitely many indicesi. These functions can equivalently be regarded asfinitely supported sections of thefiber bundle over the index setI, with the fiber over${\displaystyle i\in I}$ being${\displaystyle M_i}$.

This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing${\displaystyle (\alpha +\beta {)}_i={\alpha }_i+{\beta }_i}$ for alli (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an elementr fromR by defining${\displaystyle r(\alpha {)}_i=(r\alpha {)}_i}$ for alli. In this way, the direct sum becomes a leftR-module, and it is denoted$${\displaystyle \underset{i\in I}{⨁}{M}_i.}$$

It is customary to write the sequence${\displaystyle ({\alpha }_i)}$ as a sum${\displaystyle \sum {\alpha }_i}$. Sometimes a primed summation${\displaystyle {\sum }^{\prime }{\alpha }_i}$ is used to indicate thatcofinitely many of the terms are zero.

• The direct sum is asubmodule of thedirect product of the modulesM_i (Bourbaki 1989, §II.1.7). The direct product is the set of all functionsα fromI to the disjoint union of the modulesM_i withα(i)∈M_i, but not necessarily vanishing for all but finitely manyi. If the index setI is finite, then the direct sum and the direct product are equal.
• Each of the modulesM_i may be identified with the submodule of the direct sum consisting of those functions which vanish on all indices different fromi. With these identifications, every elementx of the direct sum can be written in one and only one way as a sum of finitely many elements from the modulesM_i.
• If theM_i are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of theM_i. The same is true for therank of abelian groups and thelength of modules.
• Every vector space over the fieldK is isomorphic to a direct sum of sufficiently many copies ofK, so in a sense only these direct sums have to be considered. This is not true for modules over arbitrary rings.
• Thetensor product distributes over direct sums in the following sense: ifN is some rightR-module, then the direct sum of the tensor products ofN withM_i (which are abelian groups) is naturally isomorphic to the tensor product ofN with the direct sum of theM_i.
• Direct sums arecommutative andassociative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum.
• The abelian group ofR-linear homomorphisms from the direct sum to some leftR-moduleL is naturally isomorphic to thedirect product of the abelian groups ofR-linear homomorphisms fromM_i toL:$${\displaystyle {\mathrm{Hom}}_R(\underset{i\in I}{⨁}{M}_i,L)\cong \prod _{i\in I}{\mathrm{Hom}}_R\left(M_i,L\right).}$$ Indeed, there is clearly ahomomorphismτ from the left hand side to the right hand side, whereτ(θ)(i) is theR-linear homomorphism sendingx∈M_i toθ(x) (using the natural inclusion ofM_i into the direct sum). The inverse of the homomorphismτ is defined by$${\displaystyle {\tau }^{-1}(\beta )(\alpha )=\sum _{i\in I}\beta (i)(\alpha (i))}$$ for anyα in the direct sum of the modulesM_i. The key point is that the definition ofτ⁻¹ makes sense becauseα(i) is zero for all but finitely manyi, and so the sum is finite.
  In particular, thedual vector space of a direct sum of vector spaces is isomorphic to thedirect product of the duals of those spaces.
• Thefinite direct sum of modules is abiproduct: If$${\displaystyle p_k:A_1\oplus \cdots \oplus A_n\to A_k}$$ are the canonical projection mappings and$${\displaystyle i_k:A_k\mapsto A_1\oplus \cdots \oplus A_n}$$ are the inclusion mappings, then$${\displaystyle i_1\circ p_1+\cdots +i_n\circ p_n}$$ equals the identity morphism ofA₁ ⊕ ⋯ ⊕A_n, and$${\displaystyle p_k\circ i_l}$$ is the identity morphism ofA_k in the casel =k, and is the zero map otherwise.

SupposeM is anR-module andM_i is asubmodule ofM for eachi inI. If everyx inM can be written in exactly one way as a sum of finitely many elements of theM_i, then we say thatM is theinternal direct sum of the submodulesM_i (Halmos 1974, §18). In this case,M is naturally isomorphic to the (external) direct sum of theM_i as defined above (Adamson 1972, p.61).

A submoduleN ofM is adirect summand ofM if there exists some other submoduleN′ ofM such thatM is theinternal direct sum ofN andN′. In this case,N andN′ are calledcomplementary submodules.

In the language ofcategory theory, the direct sum is acoproduct and hence acolimit in the category of leftR-modules, which means that it is characterized by the followinguniversal property. For everyi inI, consider thenatural embedding

${\displaystyle j_i:M_i\to \underset{i\in I}{⨁}{M}_i}$

which sends the elements ofM_i to those functions which are zero for all arguments buti. Now letM be an arbitraryR-module andf_i :M_i →M be arbitraryR-linear maps for everyi, then there exists precisely oneR-linear map

${\displaystyle f:\underset{i\in I}{⨁}{M}_i\to M}$

such thatf oj_i =f_i for alli.

The direct sum gives a collection of objects the structure of acommutativemonoid, in that the addition of objects is defined, but not subtraction. In fact, subtraction can be defined, and every commutative monoid can be extended to anabelian group. This extension is known as theGrothendieck group. The extension is done by defining equivalence classes of pairs of objects, which allows certain pairs to be treated as inverses. The construction, detailed in the article on the Grothendieck group, is "universal", in that it has theuniversal property of being unique, and homomorphic to any other embedding of a commutative monoid in an abelian group.

If the modules we are considering carry some additional structure (for example, anorm or aninner product), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain thecoproduct in the appropriatecategory of all objects carrying the additional structure. Two prominent examples occur forBanach spaces andHilbert spaces.

In some classical texts, the phrase "direct sum ofalgebras over a field" is also introduced for denoting thealgebraic structure that is presently more commonly called adirect product of algebras; that is, theCartesian product of theunderlying sets with thecomponentwise operations. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (see note below and the remark ondirect sums of rings).

A direct sum ofalgebras${\displaystyle X}$ and${\displaystyle Y}$ is the direct sum as vector spaces, with product

${\displaystyle (x_1+y_1)(x_2+y_2)=(x_1{x}_2+y_1{y}_2).}$

Consider these classical examples:

${\displaystyle \mathbf{R}\oplus \mathbf{R}}$ isring isomorphic tosplit-complex numbers, also used ininterval analysis.

${\displaystyle \mathbf{C}\oplus \mathbf{C}}$ is the algebra oftessarines introduced byJames Cockle in 1848.

${\displaystyle \mathbf{H}\oplus \mathbf{H},}$ called thesplit-biquaternions, was introduced byWilliam Kingdon Clifford in 1873.

Joseph Wedderburn exploited the concept of a direct sum of algebras in his classification ofhypercomplex numbers. See hisLectures on Matrices (1934), page 151.Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts:${\displaystyle \lambda (x\oplus y)=\lambda x\oplus \lambda y}$ while for the direct product a scalar factor may be collected alternately with the parts, but not both:${\displaystyle \lambda (x,y)=(\lambda x,y)=(x,\lambda y).}$Ian R. Porteous uses the three direct sums above, denoting them${\displaystyle {}^{2}R,{}^{2}C,{}^{2}H,}$ as rings of scalars in his analysis ofClifford Algebras and the Classical Groups (1995).

The construction described above, as well as Wedderburn's use of the termsdirect sum anddirect product follow a different convention than the one incategory theory. In categorical terms, Wedderburn'sdirect sum is acategorical product, whilst Wedderburn'sdirect product is acoproduct (or categorical sum), which (for commutative algebras) actually corresponds to thetensor product of algebras.

The direct sum of twoBanach spaces${\displaystyle X}$ and${\displaystyle Y}$ is the direct sum of${\displaystyle X}$ and${\displaystyle Y}$ considered as vector spaces, with the norm${\displaystyle \Vert (x,y)\Vert =\Vert x{\Vert }_X+\Vert y{\Vert }_Y}$ for all${\displaystyle x\in X}$ and${\displaystyle y\in Y.}$

Generally, if${\displaystyle X_i}$ is a collection of Banach spaces, where${\displaystyle i}$ traverses theindex set${\displaystyle I,}$ then the direct sum${\displaystyle \underset{i\in I}{⨁}{X}_i}$ is a module consisting of all functions${\displaystyle x}$defined over${\displaystyle I}$ such that${\displaystyle x(i)\in X_i}$ for all${\displaystyle i\in I}$ and

The norm is given by the sum above. The direct sum with this norm is again a Banach space.

For example, if we take the index set${\displaystyle I=\mathbb{N}}$ and${\displaystyle X_i=\mathbb{R},}$ then the direct sum${\displaystyle \underset{i\in \mathbb{N}}{⨁}{X}_i}$ is the space${\displaystyle {\ell }_1,}$ which consists of all the sequences${\displaystyle \left(a_i\right)}$ of reals with finite norm$\Vert a\Vert =\sum _i\left|a_i\right|.$

A closed subspace${\displaystyle A}$ of a Banach space${\displaystyle X}$ iscomplemented if there is another closed subspace${\displaystyle B}$ of${\displaystyle X}$ such that${\displaystyle X}$ is equal to the internal direct sum${\displaystyle A\oplus B.}$ Note that not every closed subspace is complemented; e.g.${\displaystyle c_0}$ is not complemented in${\displaystyle {\ell }^{\mathrm{\infty }}.}$

Let${\displaystyle \left\{\left(M_i,b_i\right):i\in I\right\}}$ be afamily indexed by${\displaystyle I}$ of modules equipped withbilinear forms. Theorthogonal direct sum is the module direct sum with bilinear form${\displaystyle B}$ defined by^[1]$${\displaystyle B\left(\left(x_i\right),\left(y_i\right)\right)=\sum _{i\in I}{b}_i\left(x_i,y_i\right)}$$in which the summation makes sense even for infinite index sets${\displaystyle I}$ because only finitely many of the terms are non-zero.

If finitely manyHilbert spaces${\displaystyle H_1,\dots ,H_n}$ are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as:$${\displaystyle ⟨\left(x_1,\dots ,x_n\right),\left(y_1,\dots ,y_n\right)⟩=⟨x_1,y_1⟩+\cdots +⟨x_n,y_n⟩.}$$

The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutuallyorthogonal subspaces.

If infinitely many Hilbert spaces${\displaystyle H_i}$ for${\displaystyle i\in I}$ are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero. However, the result will only be aninner product space and it will not necessarily becomplete. We then define the direct sum of the Hilbert spaces${\displaystyle H_i}$ to be the completion of this inner product space.

Alternatively and equivalently, one can define the direct sum of the Hilbert spaces${\displaystyle H_i}$ as the space of all functions α with domain${\displaystyle I,}$ such that${\displaystyle \alpha (i)}$ is an element of${\displaystyle H_i}$ for every${\displaystyle i\in I}$ and:

The inner product of two such function α and β is then defined as:$${\displaystyle ⟨\alpha ,\beta ⟩=\sum _i⟨{\alpha }_i,{\beta }_i⟩.}$$

This space is complete and we get a Hilbert space.

For example, if we take the index set${\displaystyle I=\mathbb{N}}$ and${\displaystyle X_i=\mathbb{R},}$ then the direct sum${\displaystyle {\oplus }_{i\in \mathbb{N}}{X}_i}$ is the space${\displaystyle {\ell }_2,}$ which consists of all the sequences${\displaystyle \left(a_i\right)}$ of reals with finite norm$\Vert a\Vert =\sqrt{\sum _i{\Vert a_i\Vert }^2}.$ Comparing this with the example forBanach spaces, we see that the Banach space direct sum and the Hilbert space direct sum are not necessarily the same. But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different.

Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field, which is either${\displaystyle \mathbb{R}\text{ or }\mathbb{C}.}$ This is equivalent to the assertion that every Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space iscomplemented because it admits anorthogonal complement. Conversely, theLindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.

• Biproduct – in category theory, an object that is both product and coproduct in compatible waysPages displaying wikidata descriptions as a fallback
• Indecomposable module
• Jordan–Hölder theorem – Decomposition of an algebraic structurePages displaying short descriptions of redirect targets
• Krull–Schmidt theorem – Mathematical theorem
• Split exact sequence – Type of short exact sequence in mathematics

• Adamson, Iain T. (1972),Elementary rings and modules, University Mathematical Texts, Oliver and Boyd,ISBN 0-05-002192-3.
• Bourbaki, Nicolas (1989),Elements of mathematics, Algebra I, Springer-Verlag,ISBN 3-540-64243-9.
• Dummit, David S.; Foote, Richard M. (1991),Abstract algebra, Englewood Cliffs, NJ: Prentice Hall, Inc.,ISBN 0-13-004771-6.
• Halmos, Paul (1974),Finite dimensional vector spaces, Springer,ISBN 0-387-90093-4
• Mac Lane, S.;Birkhoff, G. (1999),Algebra, AMS Chelsea,ISBN 0-8218-1646-2.

• Linear algebra
• Module theory

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