Direct sum

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The direct sum is an operation between
structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrou ...
in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
. The direct sum of two abelian groups $A$ and $B$ is another abelian group $A\oplus B$ consisting of the ordered pairs $\left(a,b\right)$ where $a \in A$ and $b \in B$. To add ordered pairs, we define the sum $\left(a, b\right) + \left(c, d\right)$ to be $\left(a + c, b + d\right)$; in other words addition is defined coordinate-wise. For example, the direct sum $\Reals \oplus \Reals$, where $\Reals$ is
real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the tuple, -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real v ...
, is the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry ...
, $\R ^2$. A similar process can be used to form the direct sum of two
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s or two modules. We can also form direct sums with any finite number of summands, for example $A \oplus B \oplus C$, provided $A, B,$ and $C$ 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
. That is, $\left(A \oplus B\right) \oplus C \cong A \oplus \left(B \oplus C\right)$ for any algebraic structures $A$, $B$, and $C$ of the same kind. The direct sum is also
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
up to isomorphism, i.e. $A \oplus B \cong B \oplus A$ for any algebraic structures $A$ and $B$ of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
. This 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. As an example, consider the direct sum and 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 $\left(A_i\right)_$, the direct sum $\bigoplus_ A_i$ is defined to be the set of tuples $\left(a_i\right)_$ with $a_i \in A_i$ such that $a_i=0$ for all but finitely many ''i''. The direct sum $\bigoplus_ A_i$ is contained in the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
$\prod_ A_i$, but is strictly smaller when the
index set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
$I$ is infinite, because an element of the direct product can have infinitely many nonzero coordinates.

# Examples

The ''xy''-plane, a two-dimensional
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, 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 $\left(x_1,y_1\right) + \left(x_2,y_2\right) = \left(x_1+x_2, y_1 + y_2\right)$, which is the same as vector addition. Given two structures $A$ and $B$, their direct sum is written as $A\oplus B$. Given an
indexed family In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
of structures $A_i$, indexed with $i \in I$, the direct sum may be written $A=\bigoplus_A_i$. 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 $+$ the phrase "direct sum" is used, while if the group operation is written $*$ 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 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 $\mathbb$ and then define $\mathbb \oplus \mathbb$ the direct sum is said to be external. If, on the other hand, we first define some algebraic structure $S$ and then write $S$ as a direct sum of two substructures $V$ and $W$, then the direct sum is said to be internal. In this case, each element of $S$ is expressible uniquely as an algebraic combination of an element of $V$ and an element of $W$. For an example of an internal direct sum, consider $\mathbb Z_6$ (the integers modulo six), whose elements are $\$. This is expressible as an internal direct sum $\mathbb Z_6 = \ \oplus \$.

# Types of direct sum

## Direct sum of abelian groups

The direct sum of
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
s is a prototypical example of a direct sum. Given two such groups $\left(A, \circ\right)$ and $\left(B, \bullet\right),$ their direct sum $A \oplus B$ is the same as their
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
. That is, the underlying set is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
$A \times B$ and the group operation $\,\cdot\,$ is defined component-wise: $\left(a_1, b_1\right) \cdot \left(a_2, b_2\right) = \left(a_1 \circ a_2, b_1 \bullet b_2\right).$ This definition generalizes to direct sums of finitely many abelian groups. For an arbitrary family of groups $A_i$ indexed by $i \in I,$ their $\bigoplus_ A_i$ is the
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
of the direct product that consists of the elements $\left(a_i\right)_ \in \prod_ A_i$ that have finite
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
, where by definition, $\left\left(a_i\right\right)_$ is said to have if $a_i$ is the identity element of $A_i$ for all but finitely many $i.$ The direct sum of an infinite family $\left\left(A_i\right\right)_$ of non-trivial groups is a
proper subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
of the product group $\prod_ A_i.$

## 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 space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s, which are modules over a field. The construction may also be extended to
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
s and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
s.

## Direct sum in categories

An
additive category In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition A category C is preadditive if all its hom-sets are abelian groups and composition of morp ...
is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree and the direct sum is either of them, cf.
biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of Object (category theory), objects, in a category (mathematics), category with zero objects, is both a product (category theory), product and a coproduct. ...
. General case: In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
the is often, but not always, the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of Set (mathematics), sets and disjoint union (topology), of topological spaces, the free product of Group (mathematics), group ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
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 sums versus coproducts in category of groups

However, the direct sum $S_3 \oplus \Z_2$ (defined identically to the direct sum of abelian groups) is a coproduct of the groups $S_3$ and $\Z_2$ in the
category of groups In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.

## Direct sum of group representations

The direct sum of group representations generalizes the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
of the underlying modules, adding a
group action In mathematics, a group action on a space (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
to it. Specifically, given a
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
$G$ and two
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It c ...
$V$ and $W$ of $G$ (or, more generally, two $G$-modules), the direct sum of the representations is $V \oplus W$ with the action of $g \in G$ given component-wise, that is, $g \cdot (v, w) = (g \cdot v, g \cdot w).$ Another equivalent way of defining the direct sum is as follows: Given two representations $\left(V, \rho_V\right)$ and $\left(W, \rho_W\right)$ the vector space of the direct sum is $V \oplus W$ and the homomorphism $\rho_$ is given by $\alpha \circ \left(\rho_V \times \rho_W\right),$ where $\alpha: GL\left(V\right) \times GL\left(W\right) \to GL\left(V \oplus W\right)$ is the natural map obtained by coordinate-wise action as above. Furthermore, if $V,\,W$ are finite dimensional, then, given a basis of $V,\,W$, $\rho_V$ and $\rho_W$ are matrix-valued. In this case, $\rho_$ is given as $g \mapsto \begin\rho_V(g) & 0 \\ 0 & \rho_W(g)\end.$ Moreover, if we treat $V$ and $W$ as modules over the
group ring In algebra, a group ring is a free module and at the same time a Ring (mathematics), ring, constructed in a natural way from any given ring and any given Group (mathematics), group. As a free module, its ring of scalars is the given ring, and its ...
$kG$, where $k$ is the field, then the direct sum of the representations $V$ and $W$ is equal to their direct sum as $kG$ modules.

## Direct sum of rings

Some authors will speak of the direct sum $R \oplus S$ of two rings when they mean the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
$R \times S$, but this should be avoided since $R \times S$ does not receive natural ring homomorphisms from $R$ and $S$: in particular, the map $R \to R \times S$ sending $r$ to $\left(r, 0\right)$ is not a ring homomorphism since it fails to send 1 to $\left(1, 1\right)$ (assuming that $0 \neq 1$ in $S$). Thus $R \times S$ is not a coproduct in the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
, and should not be written as a direct sum. (The coproduct in the
category of commutative rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
is the tensor product of rings., section I.11 In the category of rings, the coproduct is given by a construction similar to the
free product In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
of groups.) Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If $\left(R_i\right)_$ 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 $\mathbf$ and $\mathbf$, the direct sum $\mathbf \oplus \mathbf$ is defined as the block diagonal matrix of $\mathbf$ and $\mathbf$ if both are square matrices (and to an analogous
block matrix In mathematics, a block matrix or a partitioned matrix is a matrix (mathematics), matrix that is ''Interpretation (logic), interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block ...
, if not). $\mathbf \oplus \mathbf = \begin \mathbf & 0 \\ 0 & \mathbf \end.$

## Direct sum of topological vector spaces

A
topological vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
(TVS) $X,$ such as a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
, is said to be a of two vector subspaces $M$ and $N$ if the addition map is an isomorphism of topological vector spaces (meaning that this
linear map In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
), in which case $M$ and $N$ are said to be in $X.$ This is true if and only if when considered as
topological group In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s (so scalar multiplication is ignored), $X$ is the topological direct sum of the topological subgroups $M$ and $N.$ If this is the case and if $X$ is Hausdorff then $M$ and $N$ are necessarily closed subspaces of $X.$ If $M$ is a vector subspace of a real or complex vector space $X$ then there always exists another vector subspace $N$ of $X,$ called an such that $X$ is the of $M$ and $N$ (which happens if and only if the addition map $M \times N \to X$ 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 $M$ of $X$ is said to be a () if there exists some vector subspace $N$ of $X$ such that $X$ is the topological direct sum of $M$ and $N.$ A vector subspace is called 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
is complemented. But every
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.

# Homomorphisms

The direct sum $\bigoplus_ A_i$ comes equipped with a '' projection''
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
$\pi_j \colon \, \bigoplus_ A_i \to A_j$ for each ''j'' in ''I'' and a ''coprojection'' $\alpha_j \colon \, A_j \to \bigoplus_ A_i$ for each ''j'' in ''I''. Given another algebraic structure $B$ (with the same additional structure) and homomorphisms $g_j \colon A_j \to B$ for every ''j'' in ''I'', there is a unique homomorphism $g \colon \, \bigoplus_ A_i \to B$, called the sum of the ''g''''j'', such that $g \alpha_j =g_j$ for all ''j''. Thus the direct sum is the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of Set (mathematics), sets and disjoint union (topology), of topological spaces, the free product of Group (mathematics), group ...
in the appropriate
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
.