Editing Finitely generated module (section)

==Definition==

The left ''R''-module ''M'' is finitely generated if there exist ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> in ''M'' such that for any ''x'' in ''M'', there exist ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''n''</sub> in ''R'' with ''x'' = ''r''<sub>1</sub>''a''<sub>1</sub> + ''r''<sub>2</sub>''a''<sub>2</sub> + ... + ''r''<sub>''n''</sub>''a''<sub>''n''</sub>.

The [[Set (mathematics)|set]] {''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>} is referred to as a [[generating set of a module|generating set]] of ''M'' in this case. A finite generating set need not be a basis, since it need not be linearly independent over ''R''. What is true is: ''M'' is finitely generated if and only if there is a surjective [[module homomorphism|''R''-linear map]]:
:<math>R^n \to M</math>
for some ''n''; in other words, ''M'' is a [[Quotient module|quotient]] of a [[free module]] of finite rank.

If a set ''S'' generates a module that is finitely generated, then there is a finite generating set that is included in ''S'', since only finitely many elements in ''S'' are needed to express the generators in any finite generating set, and these finitely many elements form a generating set. However, it may occur that ''S'' does not contain any finite generating set of minimal [[cardinality]]. For example the set of the [[prime number]]s is a generating set of <math>\mathbb Z</math> viewed as <math>\mathbb Z</math>-module, and a generating set formed from prime numbers has at least two elements, while the [[singleton (mathematics)|singleton]]{{math|{{mset|1}}}} is also a generating set.

In the case where the [[module (mathematics)|module]] ''M'' is a [[vector space]] over a [[field (mathematics)|field]] ''R'', and the generating set is [[linearly independent]], ''n'' is ''well-defined'' and is referred to as the [[dimension of a vector space|dimension]] of ''M'' (''well-defined'' means that any [[linearly independent]] generating set has ''n'' elements: this is the [[dimension theorem for vector spaces]]).

Any module is the union of the [[directed set]] of its finitely generated submodules.

A module ''M'' is finitely generated if and only if any increasing chain ''M''<sub>''i''</sub> of submodules with union ''M'' stabilizes: i.e., there is some ''i'' such that ''M''<sub>''i''</sub> = ''M''. This fact with [[Zorn's lemma]] implies that every nonzero finitely generated module admits [[maximal submodule]]s. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module ''M'' is called a [[Noetherian module]].