Finitely-generated Module - Some Facts

Some Facts

Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. As an example, consider the ring R = Z of all polynomials in 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 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 over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated 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 be an exact sequence of modules. Then M is finitely generated if 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 is finitely-generated. Also, is Noetherian (resp. Artinian) if and only if are Noetherian (resp. Artinian).

Let B be a ring and A its subring such that B is a faithfully flat right A-module. Then a left A-module F is finitely generated (resp. finitely presented) if and only if the B-module is finitely generated (resp. finitely presented).