Direct sum

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The symbol  \oplus \! denotes direct sum; it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction.

The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum  \mathbf{R} \oplus \mathbf{R} , where  \mathbf{R} is the set of real numbers.  \mathbf{R} \oplus \mathbf{R} is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.

To see how direct sum is used in abstract algebra, consider the most 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 or all rings or all vector spaces.

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 sum 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[edit]

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 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. In the direct sum, all but finitely many coordinates must be zero.

Internal and external direct sums[edit]

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 R and then define R \oplus R the direct sum is said to be external. If, on the other hand, we first define some set, S and then write S as the direct sum of two of its proper subsets, then the direct sum is said to be internal. For an example of an internal direct sum, consider Z_6, the integers modulo six, whose elements are \{0, 1, 2, 3, 4, 5\}.  Z_6 =\{0, 3\} \oplus \{0, 2, 4\}.

Types of direct sum[edit]

Direct sum of abelian groups[edit]

The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups (A,\ast) and (B, \cdot) , 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  \circ is defined component-wise:

(a_1, b_1) \circ (a_2, b_2) = (a_1 \ast a_2, b_1 \cdot b_2).

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

For an infinite family of abelian groups Ai for iI, 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 ai is the identity element of Ai for all but finitely many i.[2]

Direct sum of modules[edit]

Main article: 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[edit]

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 VW with the action of gG given component-wise, i.e.

g·(v, w) = (g·v, g·w).

Direct sum of rings[edit]

Main article: Product 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[edit]

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[edit]

The direct sum \bigoplus_{i \in I} A_i comes equipped with a homomorphism \alpha_j \colon A_j \to \bigoplus_{i \in I} A_i for each j. Given another abelian group B (with the same additional structure) equipped with a homomorphism 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 gj) such that g \alpha_j =g_j for all j. Thus the direct sum is the coproduct in the appropriate category.

See also[edit]

Notes[edit]

  1. ^ Thomas W. Hungerford, Algebra, p.60, Springer, 1974, ISBN 0387905189
  2. ^ Joseph J. Rotman, The Theory of Groups: an Introduction, p. 177, Allyn and Bacon, 1965
  3. ^ Math StackExchange on direct sum of rings vs. direct product of rings.
  4. ^ Lang 2002, section I.11
  5. ^ "p.45"
  6. ^ "appendix"
  7. ^ [1]

References[edit]