direct sum について

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 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 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 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)_

, 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 product

\prod_ 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.〔Thomas W. Hungerford, ''Algebra'', p.60, Springer, 1974, ISBN 0387905189〕
==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_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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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