Table of Contents - 11.1.1. Direct sum of left modules

According to Wikipedia ("Direct sum of modules", 28-Mar-2019,
https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and
{ M<sub>i</sub>: i ∈ I } a family of left R-modules indexed by the set I.
The direct sum of {M<sub>i</sub>} is then defined to be the set of all
sequences (α<sub>i</sub>) where α<sub>i</sub> ∈ M<sub>i</sub>
and α<sub>i</sub> = 0 for cofinitely many indices i. (The direct product
is analogous but the indices do not need to cofinitely vanish.)". In this
definition, "cofinitely many" means "almost all" or "for all but finitely
many". Furthemore, "This set inherits the module structure via componentwise
addition and scalar multiplication. Explicitly, two such sequences α and
β can be added by writing (α + β)<sub>i</sub> =
α<sub>i</sub> + β<sub>i</sub> for all i (note that this is again
zero for all but finitely many indices), and such a sequence can be multiplied
with an element r from R by defining r(α)<sub>i</sub> =
(rα)<sub>i</sub> for all i.".
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In [Lang] p. 128, the definition of the direct sum of left modules
is based on direct sums of abelian groups ("We define on [the direct sum of
abelian groups M<sub>i</sub>] M a structure of A-module: If
(x<sub>i</sub>)<sub>i ∈ I</sub> is an element of M, i.e. a familiy of
elements x<sub>i</sub> ∈ M<sub>i</sub> such that x<sub>i</sub> = 0 for
almost all i, and if a ∈ A, then we define a(x<sub>i</sub>)<sub>i ∈
I</sub> = (ax<sub>i</sub>)<sub>i ∈ I</sub>, that is we define
multiplication by a componentwise.") which itself is based on the direct
product of abelian groups ([Lang] p. 36: "Let {A<sub>i</sub>}<sub>i ∈
I</sub> be a family of abelian groups. We define their direct sum A ... to be
the subset of the direct product ... consisting of all families
(x<sub>i</sub>)<sub>i ∈ I</sub> with x<sub>i</sub> ∈ A<sub>i</sub>
such that x<sub>i</sub> = 0 for all but a finite number of indices i").
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In short, the <b>direct sum</b> of a familiy of (left) modules
{M<sub>i</sub>}<sub>i ∈ I</sub> is the restriction of the direct product
of {M<sub>i</sub>}<sub>i ∈ I</sub> to the elements with index function
having finite support, as formalized by the definition df-dsmm.