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Some authors will speak of the direct sum ${\displaystyle R\oplus S}$ of two rings when they mean the direct product ${\displaystyle R\times S}$, but this should be avoided[3] since ${\displaystyle R\times S}$ does not receive natural ring homomorphisms from R and S: in particular, the map ${\displaystyle R\to R\times S}$ sending r to (r,0) is not a ring homomorphism since it fails to send 1 to (1,1) (assuming that 0≠1 in S). Thus ${\displaystyle R\times S}$ is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings.[4] In the category of rings, the coproduct is given by a construction similar to the free product of groups.)

Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If

Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If ${\displaystyle (R_{i})_{i\in I}}$ is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, i.e., a ring without a multiplicative identity.

An additive category is an abstraction of the properties of the category of modules.[5][6] In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case:[7] In category theory the direct sum is often, but not always, the coproduct in the [7] In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.

The direct sum