Editing Finitely generated module (section)

==Some facts==

Every [[module homomorphism|homomorphic image]] of a finitely generated module is finitely generated. In general, [[submodule]]s of finitely generated modules need not be finitely generated. As an example, consider the ring ''R''&nbsp;=&nbsp;'''Z'''[''X''<sub>1</sub>, ''X''<sub>2</sub>, ...] of all [[polynomial]]s in [[countable|countably many]] variables. ''R'' itself is a finitely generated ''R''-module (with {1} as generating set). Consider the submodule ''K'' consisting of all those polynomials with zero constant term. Since every polynomial contains only finitely many terms whose coefficients are non-zero, the ''R''-module ''K'' is not finitely generated.

In general, a module is said to be [[noetherian module|Noetherian]] if every submodule is finitely generated. A finitely generated module over a [[Noetherian ring]] is a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This resembles, but is not exactly  [[Hilbert's basis theorem]], which states that the polynomial ring ''R''[''X''] over a Noetherian ring ''R'' is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.

More generally, an algebra (e.g., ring) that is a finitely generated module is a [[finitely generated algebra]]. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See [[integral element]] for more.)

Let 0 → ''M''′ → ''M'' → ''M''′′ → 0 be an [[exact sequence]] of modules. Then ''M'' is finitely generated if ''M''′, ''M''′′ are finitely generated. There are some partial converses to this. If ''M'' is finitely generated and ''M''′′ is finitely presented (which is stronger than finitely generated; see below), then ''M''′ is finitely generated. Also, ''M'' is Noetherian (resp. Artinian) if and only if ''M''′, ''M''′′ are Noetherian (resp. Artinian).

Let ''B'' be a ring and ''A'' its subring such that ''B'' is a [[faithfully flat module|faithfully flat]] right ''A''-module. Then a left ''A''-module ''F'' is finitely generated (resp. finitely presented) if and only if the ''B''-module {{nowrap|''B'' ⊗<sub>''A''</sub> ''F''}} is finitely generated (resp. finitely presented).{{sfn|Bourbaki|1998|loc=Ch 1, §3, no. 6, Proposition 11}}