Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module, finite over R, or a module of finite type.
Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
The left R-module M is finitely generated if there exist a1, a2, ..., an in M such that for all x in M, there exist r1, r2, ..., rn in R with x = r1a1 + r2a2 + ... + rnan.
The set {a1, a2, ..., an} is referred to as a generating set for M in this case. The finite generators need not be a basis, since they need not be linearly independent over R. What is true is: M is finitely generated if and only if there is a surjective R-linear map:
for some n (M is a quotient of a free module of finite rank.)
If a set S generates a module that is finitely generated, then the finite generators of the module can be taken from S at the expense of possibly increasing the number of the generators (since only finitely many elements in S are needed to express the finite generators).
In the case where the module M is a vector space over a field R, and the generating set is linearly independent, n is well-defined and is referred to as the dimension of M (well-defined means that any linearly independent generating set has n elements: this is the dimension theorem for vector spaces).
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