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
, where
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 ...

,
. 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
and
is another abelian group
consisting of the ordered pairs
where
and
with the following structure. To add ordered pairs, we define the sum
to be
; 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
, provided
and
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 ...

. That is,
for any algebraic structures
,
, and
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.
for any algebraic structures
and
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
, the direct sum
is defined to be the set of tuples
with
such that
for all but finitely many ''i''. The direct sum
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 ...
, 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 ...
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
, which is the same as vector addition.
Given two structures
and
, their direct sum is written as
. 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
, indexed with
, the direct sum may be written
. Each ''A
i'' 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
and then define
the direct sum is said to be external.
If, on the other hand, we first define some algebraic structure
and then write
as a direct sum of two substructures
and
, then the direct sum is said to be internal. In this case, each element of
is expressible uniquely as an algebraic combination of an element of
and an element of
. For an example of an internal direct sum, consider
(the integers modulo six), whose elements are
. This is expressible as an internal direct sum
.
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 ...
and
their direct sum
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 ...
and the group operation
is defined component-wise:
This definition generalizes to direct sums of finitely many abelian groups.
For an arbitrary family of groups
indexed by
their
[
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 that have finite support, where by definition, is said to have if is the identity element of for all but finitely many
The direct sum of an infinite family 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
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 (defined identically to the direct sum of abelian groups) is a coproduct of the groups and 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 ...
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 ...
and of (or, more generally, two -modules), the direct sum of the representations is with the action of given component-wise, that is,
Another equivalent way of defining the direct sum is as follows:
Given two representations and the vector space of the direct sum is and the homomorphism is given by where is the natural map obtained by coordinate-wise action as above.
Furthermore, if are finite dimensional, then, given a basis of , and are matrix-valued. In this case, is given as
Moreover, if we treat and 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 ...
, where is the field, then the direct sum of the representations and is equal to their direct sum as modules.
Direct sum of rings
Some authors will speak of the direct sum 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 ...
, but this should be avoided since does not receive natural ring homomorphisms from and : in particular, the map sending to is not a ring homomorphism since it fails to send 1 to (assuming that in ). Thus 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 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 the direct sum 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 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).
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) 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 and if the addition map
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 ...

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 ...

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 and are said to be in
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), is the topological direct sum of the topological subgroups and
If this is the case and if is then and are necessarily closed subspaces of
If is a vector subspace of a real or complex vector space then there always exists another vector subspace of called an such that is the of and (which happens if and only if the addition map 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 of is said to be a () if there exists some vector subspace of such that is the topological direct sum of and
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 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 ...
for each ''j'' in ''I'' and a ''coprojection'' for each ''j'' in ''I''. Given another algebraic structure (with the same additional structure) and homomorphisms for every ''j'' in ''I'', there is a unique homomorphism , called the sum of the ''g''''j'', such that 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