Direct sum

The direct sum is a construction of a structure from a set of smaller structures with identity elements. The construction can be applied to groups, rings, and modules.

Specifically, let $(A_i)_{i\in I}$ be a family of structures of the same species with identity elements. The direct sum of the family $(A_i)_{i\in I}$, denoted \[\bigoplus_{i\in I} A_i,\] is the subset of the direct product of the $A_i$ of elements $(x_i)_{i\in I}$ with only finitely many non-identity coordinates. Note the difference between this product and the direct product the two structures coincide when the family $(A_i)_{i\in I}$ is finite, or when all but finitely many of the $A_i$ are trivial structures (i.e., they consist of the identity alone).

When finitely many structures are involved, the terms direct product and direct sum are interchangeable, but in the case of groups, direct sum is normally used for abelian groups, since additive notation is usually used for a commutative operation.

The direct sum of a family of groups is a normal subgroup of the direct product of the family of groups. For if $a$ is an element of the direct product, and $x$ is an element of the direct sum, then all but finitely many of the coordinates of the element $xax^{-1}$ are the same as the coordinates of $xx^{-1}=e$.

See also