Direct sum

In mathematics, more specifically in algebra, the direct sum of a collection of abelian groups is an abelian group constructed by combining the given groups as described below. If the input abelian groups have additional structure (for example, are vector spaces, modules, or topological abelian groups), then the direct sum usually maintains that structure (as an example of the opposite, the direct sum of fields is not a field as it contains zero divisors).

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}

. The sum

(

a

,

b

)

+

(

c

,

d

)

{\displaystyle (a,b)+(c,d)}

is defined to be

(

a

+

c

,

b

+

d

)

{\displaystyle (a+c,b+d)}

; in other words, addition is defined coordinate-wise and it is usually denoted with

+

{\displaystyle +}

. For example, the direct sum

R

⊕

R

{\displaystyle \mathbb {R} \oplus \mathbb {R} }

, where

R

{\displaystyle \mathbb {R} }

is real coordinate space, is the Cartesian plane,

R

2

{\displaystyle \mathbb {R} ^{2}}

.

Direct sums can also be formed 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). That 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.