In
mathematics, a finitely generated module is a
module that has a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
generating set. A finitely generated module over a
ring ''R'' may also 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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, and a finitely generated module over the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is simply a
finitely generated abelian group.
Definition
The left ''R''-module ''M'' is finitely generated if there exist ''a''
1, ''a''
2, ..., ''a''
''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''
1, ''r''
2, ..., ''r''
''n'' in ''R'' with ''x'' = ''r''
1''a''
1 + ''r''
2''a''
2 + ... + ''r''
''n''''a''
''n''.
The
set is referred to as a
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