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In [[abstract algebra|algebra]], a '''generating set''' ''G'' of a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is a subset of ''M'' such that the smallest submodule of ''M'' containing ''G'' is ''M'' itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set ''G'' is then said to generate ''M''. For example, the ring ''R'' is generated by the identity element 1 as a left ''R''-module over itself. If there is a finite generating set, then a module is said to be [[finitely generated module|finitely generated]].
Explicitly, if ''G'' is a generating set of a module ''M'', then every element of ''M'' is a (finite) ''R''-linear combination of some elements of ''G''; i.e., for each ''x'' in ''M'', there are ''r''<sub>1</sub>, ..., ''r''<sub>''m''</sub> in ''R'' and ''g''<sub>1</sub>, ..., ''g''<sub>''m''</sub> in ''G'' such that
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Put in another way, there is a surjection
: <math> \bigoplus_{g \in G} R \to M, \, r_g \mapsto r_g g
where we wrote ''r''<sub>''g''</sub> for an element in the ''g''-th component of the direct sum. (Coincidentally, since a generating set always exists; for example, ''M'' itself, this shows that a module is a quotient of a free module, a useful fact.)
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