Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum $\mathbf{R} \oplus \mathbf{R}$ , where $\mathbf{R}$ is real coordinate space, is the Cartesian plane, $\mathbf {R} ^{2}$ . To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups $A$ and $B$ is another abelian group $A\oplus B$ consisting of the ordered pairs $(a,b)$ where $a\in A$ and $b\in B$ . 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 any two algebraic structures, such as rings, modules, and vector spaces.

We can also form direct sums with any number of summands, for example $A \oplus B \oplus C$ , provided $A, B,$ and $C$ are the same kinds of algebraic structures, that is, all groups, rings, vector spaces, etc.

In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression $xy$ ) we use direct product.

In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be 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. More generally, 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 $(A_i)_{i \in I}$ , the direct sum $\bigoplus_{i \in I} A_i$ is defined to be the set of tuples $(a_i)_{i \in I}$ with $a_i \in A_i$ such that $a_i=0$ for all but finitely many i. The direct sum $\bigoplus_{i \in I} A_i$ is contained in the direct product $\prod_{i \in I} A_i$ , but is usually strictly smaller when the index set $I$ is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.^[1]

Examples

For example, the xy-plane, a two-dimensional vector space, 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 objects $A$ and $B$ , their direct sum is written as $A\oplus B$ . Given an indexed family of objects $A_{i}$ , indexed with $i\in I$ , the direct sum may be written $\textstyle A=\bigoplus_{i\in I}A_i$ . 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. In the direct sum, 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 $\mathbf {R}$ and then define $\mathbf{R} \oplus \mathbf{R}$ the direct sum is said to be external.

If, on the other hand, we first define some algebraic object, $S$ and then write $S$ as the direct sum of two of its proper subsets, $V$ 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 $Z_6$ , the integers modulo six, whose elements are $\{0, 1, 2, 3, 4, 5\}$ . This is expressible as an internal direct sum $Z_6 =\{0, 3\} \oplus \{0, 2, 4\}$ .

Types of direct sum

Direct sum of abelian groups

The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups $(A,\circ )$ and $(B,\bullet )$ , their direct sum $A\oplus B$ is the same as their direct product, that is the underlying set is the Cartesian product $A\times B$ and the group operation $\cdot$ is defined component-wise:

(a_{1},b_{1})\cdot (a_{2},b_{2})=(a_{1}\circ a_{2},b_{1}\bullet b_{2})

This definition generalizes to direct sums of finitely many abelian groups.

For an infinite family of abelian groups A_i for i ∈ I, the direct sum

\bigoplus _{i\in I}A_{i}

is a proper subgroup of the direct product. It consists of the elements $\textstyle (a_{i})\in \prod _{j\in I}A_{j}$ such that a_i is the identity element of A_i for all but finitely many i.^[2]

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 spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.

Direct sum of group representations

The direct sum of group representations generalizes the direct sum 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.

g·(v, w) = (g·v, g·w).

Direct sum of rings

Some authors will speak of the direct sum $R \oplus S$ of two rings when they mean the direct product $R \times S$ , but this should be avoided^[3] 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≠1 in S). Thus $R \times S$ is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings.^[4] In the category of rings, the coproduct is given by a construction similar to the free product of groups.)

Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If $(R_i)_{i \in I}$ 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, i.e., a ring without a multiplicative identity.

Direct sum in categories

An additive category is an abstraction of the properties of the category of modules.^[5] ^[6]

In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case : ^[7] In category theory the direct sum is often, but not always, the coproduct in the category 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.

Homomorphisms

The direct sum $\bigoplus_{i \in I} A_i$ comes equipped with a projection homomorphism $\pi_j \colon \bigoplus_{i \in I} A_i \to A_j$ for each j and a coprojection $\alpha_j \colon A_j \to \bigoplus_{i \in I} A_i$ for each j.^[8] Given another algebraic object B (with the same additional structure) and homomorphisms $g_j \colon A_j \to B$ for every j, there is a unique homomorphism $g \colon \bigoplus_{i \in I} A_i \to B$ (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 the appropriate category.

Notes

↑ Thomas W. Hungerford, Algebra, p.60, Springer, 1974, ISBN 0387905189
↑ Joseph J. Rotman, The Theory of Groups: an Introduction, p. 177, Allyn and Bacon, 1965
↑ Math StackExchange on direct sum of rings vs. direct product of rings.
↑ Lang 2002, section I.11
↑ "p.45"
↑ "appendix"
↑
↑ Heunen, Chris (2009). Categorical Quantum Models and Logics. Pallas Proefschriften. Amsterdam University Press. p. 26. ISBN 9085550246.

References

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001

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