Editing Direct sum of modules (section)

== Internal direct sum ==
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{{see also|Internal direct product}}
Suppose ''M'' is an ''R''-module and ''M''<sub>''i''</sub> is a [[submodule]] of ''M'' for each ''i'' in ''I''. If every ''x'' in ''M'' can be written in exactly one way as a sum of finitely many elements of the ''M''<sub>''i''</sub>, then we say that ''M'' is the '''internal direct sum''' of the submodules ''M''<sub>''i''</sub> {{harv|Halmos|1974|loc=§18}}. In this case, ''M'' is naturally isomorphic to the (external) direct sum of the ''M''<sub>''i''</sub> as defined above {{harv|Adamson|1972|loc=p.61}}.

A submodule ''N'' of ''M'' is a '''direct summand''' of ''M'' if there exists some other submodule ''N′'' of ''M'' such that ''M'' is the ''internal'' direct sum of ''N'' and ''N′''. In this case, ''N'' and ''N′'' are called '''complementary submodules'''.