Editing Direct sum of modules (section)

== Construction for an arbitrary family of modules ==

One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two [[module (mathematics)|modules]]. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows {{harv|Bourbaki|1989|loc=§II.1.6}}.

Let ''R'' be a ring, and {''M''<sub>''i''</sub>&nbsp;:&nbsp;''i''&nbsp;∈&nbsp;''I''} a [[indexed family|family]] of left ''R''-modules indexed by the [[Set (mathematics)|set]] ''I''. The ''direct sum'' of {''M''<sub>''i''</sub>} is then defined to be the set of all sequences <math>(\alpha_i)</math> where <math>\alpha_i \in M_i</math> and <math>\alpha_i = 0</math> for [[cofinitely many]] indices ''i''. (The [[direct product]] is analogous but the indices do not need to cofinitely vanish.)

It can also be defined as [[function (mathematics)|functions]] α from ''I'' to the [[disjoint union]] of the modules ''M''<sub>''i''</sub> such that α(''i'')&nbsp;∈&nbsp;''M''<sub>''i''</sub> for all ''i'' ∈ ''I'' and α(''i'') = 0 for [[cofinitely many]] indices ''i''.  These functions can equivalently be regarded as [[compact support|finitely supported]] sections of the [[fiber bundle]] over the index set ''I'', with the fiber over <math>i \in I</math> being <math>M_i</math>.

This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing <math>(\alpha + \beta)_i = \alpha_i + \beta_i</math> for all ''i'' (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element ''r'' from ''R'' by defining <math>r(\alpha)_i = (r\alpha)_i</math> for all ''i''. In this way, the direct sum becomes a left ''R''-module, and it is denoted
<math display=block>\bigoplus_{i \in I} M_i.</math>

It is customary to write the sequence <math>(\alpha_i)</math> as a sum <math> \sum \alpha_i</math>. Sometimes a primed summation <math> \sum ' \alpha_i</math> is used to indicate that [[cofinitely many]] of the terms are zero.