Editing Finitely generated module (section)

==<span id="Finitely presented module"></span>Finitely presented, finitely related, and coherent modules==

Another formulation is this: a finitely generated module ''M'' is one for which there is an [[epimorphism]] mapping ''R<sup>k</sup>'' onto ''M'' :

:f : ''R<sup>k</sup>'' → ''M''.

Suppose now there is an epimorphism,

:''φ'' : ''F'' → ''M''.

for a module ''M'' and free module ''F''.

* If the [[kernel (algebra)|kernel]] of ''φ'' is finitely generated, then ''M'' is called a '''finitely related module'''.  Since ''M'' is isomorphic to ''F''/ker(''φ''), this basically expresses that ''M'' is obtained by taking a free module and introducing finitely many relations within ''F'' (the generators of ker(''φ'')).
* If the kernel of ''φ'' is finitely generated and ''F'' has finite rank (i.e. {{nowrap|1=''F'' = ''R''<sup>''k''</sup>}}), then ''M'' is said to be a '''finitely presented module'''. Here, ''M'' is specified using finitely many generators (the images of the ''k'' generators of {{nowrap|1=''F'' = ''R''<sup>''k''</sup>}}) and finitely many relations (the generators of ker(''φ'')). See also: [[free presentation]]. Finitely presented modules can be characterized by an abstract property within the [[category of modules|category of ''R''-modules]]: they are precisely the [[compact object (mathematics)|compact objects]] in this category.
*A '''coherent module''' ''M'' is a finitely generated module whose finitely generated submodules are finitely presented.

Over any ring ''R'', coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a [[Noetherian ring]] ''R'', finitely generated, finitely presented, and coherent are equivalent conditions on a module.

Some crossover occurs for projective or flat modules.  A finitely generated projective module is finitely presented, and a finitely related flat module is projective.

It is true also that the following conditions are equivalent for a ring ''R'':
# ''R'' is a right [[coherent ring]].
# The module ''R''<sub>''R''</sub> is a coherent module.
# Every finitely presented right ''R'' module is coherent.

Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since the [[category (mathematics)|category]] of coherent modules is an [[abelian category]], while, in general, neither finitely generated nor finitely presented modules form an abelian category.