The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups (A, ∗) and (B, ·), their direct sum A ⊕ B is the same as their direct product, i.e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise:
- (a1, b1) ○ (a2, b2) = (a1 ∗ a2, b1 · b2).
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups Ai for i ∈ I, the direct sum
is a proper subgroup of the direct product. It consists of the elements such that ai is the identity element of Ai for all but finitely many i.
In this case, the direct sum is indeed the coproduct in the category of abelian groups.
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