Direct sum
Encyclopedia
In mathematics
, one can often define a direct sum of objects
already known, giving a new one. This is generally the Cartesian product
of the underlying sets (or some subset
of it), together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct
in the category
in question. In cases where an object is expressed as a direct sum of subobjects, the direct sum can be referred to as an internal direct sum.
The direct sum of a family of objects Ai, with i ∈ I, is denoted by
and each Ai is called a direct summand of A.
Examples include the direct sum of abelian groups, the direct sum of modules
, the direct sum of rings, the direct sum of matrices, and the direct sum of topological spaces.
A related concept is that of the direct product
, which is sometimes the same as the direct sum, but at other times can be entirely different.
, i.e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise: ○ (a2, b2) = (a1 ∗ a2, b1 · b2).
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups Ai for i ∈ I, the direct sum
is a proper subgroup of the direct product. It consists of the elements
such that ai is the identity element of Ai for all but finitely many i.
In this case, the direct sum is indeed the coproduct in the category of abelian groups
.
of the underlying modules
, adding a group action
to it. Specifically, given a group
G and two representations
V and W of G (or, more generally, two G-modules
), the direct sum of the representations is V ⊕ W with the action of g ∈ G given component-wise, i.e.
of the Ri is sometimes called the direct sum.
Note that in the category of commutative rings, the direct sum is not the coproduct. Instead, the coproduct is the tensor product of rings.
For example, the real vector space R2 = {(x, y) : x, y ∈ R} is the direct sum of the x-axis {(x, 0) : x ∈ R} and the y-axis {(0, y) : y ∈ R}, and the sum of (x, 0) and (0, y) is the "internal" sum in the vector space R2; thus, this is an internal direct sum. More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if
In other words, every element of V can be written uniquely as the sum of an element in U with an element of W
Another case is that of abelian groups. For example, the Klein four-group
V = {e, a, b, ab} is the (internal) direct sum of the cyclic subgroups <a> and <b>.
By contrast, a direct sum of two objects which are not subobjects of a common object is an . Note however that "external direct sum" is also used to refer to an infinite direct sum of groups, to contrast with the (larger) direct product
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, one can often define a direct sum of objects
already known, giving a new one. This is generally the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the underlying sets (or some subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of it), together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
in the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
in question. In cases where an object is expressed as a direct sum of subobjects, the direct sum can be referred to as an internal direct sum.
The direct sum of a family of objects Ai, with i ∈ I, is denoted by
and each Ai is called a direct summand of A.Examples include the direct sum of abelian groups, the direct sum of modules
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
, the direct sum of rings, the direct sum of matrices, and the direct sum of topological spaces.
A related concept is that of the direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
, which is sometimes the same as the direct sum, but at other times can be entirely different.
Direct sum of abelian groups
The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups (A, ∗) and (B, ·), their direct sum A ⊕ B is the same as their direct productDirect product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
, i.e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise: ○ (a2, b2) = (a1 ∗ a2, b1 · b2).
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups Ai for i ∈ I, the direct sum
is a proper subgroup of the direct product. It consists of the elements
such that ai is the identity element of Ai for all but finitely many i.In this case, the direct sum is indeed the coproduct in the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
.
Group representations
The direct sum of group representations generalizes the direct sumDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of the underlying modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, adding a group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
to it. Specifically, given a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G and two representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
V and W of G (or, more generally, two G-modules
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
), the direct sum of the representations is V ⊕ W with the action of g ∈ G given component-wise, i.e.
- g·(v, w) = (g·v, g·w).
Direct sum of rings
Given a finite family of rings R1, ..., Rn, the direct productProduct of rings
In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
of the Ri is sometimes called the direct sum.
Note that in the category of commutative rings, the direct sum is not the coproduct. Instead, the coproduct is the tensor product of rings.
Internal direct sum
An internal direct sum is simply a direct sum of subobjects of an object.For example, the real vector space R2 = {(x, y) : x, y ∈ R} is the direct sum of the x-axis {(x, 0) : x ∈ R} and the y-axis {(0, y) : y ∈ R}, and the sum of (x, 0) and (0, y) is the "internal" sum in the vector space R2; thus, this is an internal direct sum. More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if
- U + W = {u + w : u ∈ U, w ∈ W} = V, and
- if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0.
In other words, every element of V can be written uniquely as the sum of an element in U with an element of W
Another case is that of abelian groups. For example, the Klein four-group
Klein four-group
In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
V = {e, a, b, ab} is the (internal) direct sum of the cyclic subgroups <a> and <b>.
By contrast, a direct sum of two objects which are not subobjects of a common object is an . Note however that "external direct sum" is also used to refer to an infinite direct sum of groups, to contrast with the (larger) direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
.