The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum
R
⊕
R
{\displaystyle \mathbf {R} \oplus \mathbf {R} }
, where
R
{\displaystyle \mathbf {R} }
is real coordinate space, is the Cartesian plane,
R
2
{\displaystyle \mathbf {R} ^{2}}
. To see how the 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
{\displaystyle A}
and
B
{\displaystyle B}
is another abelian group
A
⊕
B
{\displaystyle A\oplus B}
consisting of the ordered pairs
(
a
,
b
)
{\displaystyle (a,b)}
where
a
∈
A
{\displaystyle a\in A}
and
b
∈
B
{\displaystyle b\in B}
. (Confusingly this ordered pair is also called the cartesian product of the two groups.) To add ordered pairs, we define the sum
(
a
,
b
)
+
(
c
,
d
)
{\displaystyle (a,b)+(c,d)}
to be
(
a
+
c
,
b
+
d
)
{\displaystyle (a+c,b+d)}
; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of two vector spaces or two modules.
We can also form direct sums with any finite number of summands, for example
A
⊕
B
⊕
C
{\displaystyle A\oplus B\oplus C}
, provided
A
,
B
,
{\displaystyle A,B,}
and
C
{\displaystyle C}
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 up to isomorphism. That is,
(
A
⊕
B
)
⊕
C
≅
A
⊕
(
B
⊕
C
)
{\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}
for any algebraic structures
A
{\displaystyle A}
,
B
{\displaystyle B}
, and
C
{\displaystyle C}
of the same kind. The direct sum is also commutative up to isomorphism, i.e.
A
⊕
B
≅
B
⊕
A
{\displaystyle A\oplus B\cong B\oplus A}
for any algebraic structures
A
{\displaystyle A}
and
B
{\displaystyle B}
of the same kind.
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
x
y
{\displaystyle 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
∈
I
{\displaystyle (A_{i})_{i\in I}}
, the direct sum
⨁
i
∈
I
A
i
{\displaystyle \bigoplus _{i\in I}A_{i}}
is defined to be the set of tuples
(
a
i
)
i
∈
I
{\displaystyle (a_{i})_{i\in I}}
with
a
i
∈
A
i
{\displaystyle a_{i}\in A_{i}}
such that
a
i
=
0
{\displaystyle a_{i}=0}
for all but finitely many i. The direct sum
⨁
i
∈
I
A
i
{\displaystyle \bigoplus _{i\in I}A_{i}}
is contained in the direct product
∏
i
∈
I
A
i
{\displaystyle \prod _{i\in I}A_{i}}
, but is usually strictly smaller when the index set
I
{\displaystyle I}
is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.