Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If 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. In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.
In category theory the direct sum is often, but not always, the coproduct in the 
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