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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, e.g. ''M'' itself, this shows that a module is a [[quotient module|quotient]] of a [[free module]], a useful fact.)
A generating set of a module is said to be '''minimal''' if no [[proper subset]] of the set generates the module. If ''R'' is a [[field (mathematics)|field]], then a minimal generating set is the same thing as a [[basis (linear algebra)|basis]]. Unless the module is [[finitely
The [[cardinality]] of a minimal generating set need not be an invariant of the module; '''Z''' is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set {{nowrap|{2, 3}}}. What ''is'' uniquely determined by a module is the [[infimum]] of the numbers of the generators of the module.
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