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The direct sum is an operation from
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. For example, the direct sum \mathbf \oplus \mathbf , where \mathbf is
real coordinate space of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification ...
, is the
Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian plane
, \mathbf ^2 . To see how the direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The direct sum of two
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B with the following structure. To add ordered pairs, we define the sum (a, b) + (c, d) to be (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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s or two
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
. 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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
. That is, (A \oplus B) \oplus C \cong A \oplus (B \oplus C) for any algebraic structures A, B, and C of the same kind. The direct sum is also
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. 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 (A_i)_, the direct sum \bigoplus_ A_i is defined to be the set of tuples (a_i)_ 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 productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
\prod_ A_i, but is strictly smaller when the
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
I is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.


Examples

The ''xy''-plane, a two-dimensional
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, 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 (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 A and B, their direct sum is written as A\oplus B. Given an
indexed family In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of structures A_i, indexed with i \in I, the direct sum may be written \textstyle 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 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 \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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s is a prototypical example of a direct sum. Given two such
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
(A, \circ) and (B, \bullet), their direct sum A \oplus B is the same as their
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. That is, the underlying set is the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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, where by definition, \left(a_i\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(A_i\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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
into a new module. The most familiar examples of this construction occur when considering
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, which are modules over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. The construction may also be extended to
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s and
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s.


Direct sum in categories

An
additive category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
the is often, but not always, the
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of the underlying
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
, adding a
group action In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to it. Specifically, given a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 co ...
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 (V, \rho_V) and (W, \rho_W) the vector space of the direct sum is V \oplus W and the homomorphism \rho_ is given by \alpha \circ (\rho_V \times \rho_W), where \alpha: GL(V) \times GL(W) \to GL(V \oplus W) 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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 (r, 0) is not a ring homomorphism since it fails to send 1 to (1, 1) (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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is the
tensor product of rings In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
., section I.11 In the category of rings, the coproduct is given by a construction similar to the
free product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of groups.) Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If (R_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 \mathbf \text \mathbf, the direct sum \mathbf \oplus \mathbf is defined as the
block diagonal 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 m ...
of \mathbf \text \mathbf if both are square matrices (and to an analogous
block matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, if not). \mathbf \oplus \mathbf = \begin \mathbf & 0 \\ 0 & \mathbf \end.


Direct sum of topological vector spaces

A
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
(TVS) X, such as a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
for example, is said to be a of two vector subspaces M and N if the addition map \begin \ \;&& M \times N &&\;\to \;& X \\ .3ex && (m, n) &&\;\mapsto\;& m + n \\ \end is an isomorphism of topological vector spaces (meaning that this
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
is a
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijective
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
), in which case M and N are said to be in X. This is true if and only if when considered as
additive Additive may refer to: Mathematics * Additive function In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer ''n'' such that whenever ''a'' and ''b'' are coprime, the function of the product is the ...
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is complemented. But every
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
\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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the appropriate
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
.


See also

*
Direct sum of groups DIRECT was a late-2000s proposed alternative super heavy lift launch vehicle architecture supporting NASA's Vision for Space Exploration that would replace the space agency's planned Ares I and Ares V rockets with a family of Shuttle-Derived Launch ...
*
Direct sum of permutationsIn combinatorics, the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation ''π'' of length ''m'' and the permutation ''σ'' of length ''n'', the skew sum of ''π'' and ''σ ...
*
Direct sum of topological groupsIn mathematics, a topological group ''G'' is called the topological direct sum of two subgroups ''H''1 and ''H''2 if the map :\begin H_1\times H_2 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end is a topological isomorphism. More ...
*
Restricted product In mathematics, the restricted product is a construction in the theory of topological groups. Let I be an index set; S a Finite set, finite subset of I. If G_i is a locally compact group for each i \in I, and K_i \subset G_i is an open Compact grou ...
*
Whitney sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...


Notes


References

*{{Lang Algebra, edition=3r Abstract algebra