Noetherianness is about making sure small modules don’t have big parts. But how do you measure how big a module is? Remember that while every vector space has a dimension—the size of a basis—most modules don’t have a basis. We can still talk about a generating set for an -module , though, like for the -module : a subset of for which every element can be written as an -linear combination of those generating elements.
Question: For a vector space, its dimension is its minimum number of generators. Can we use the minimum number of generators as a measure of the size of a module in general?
Answer: Not really. You’d expect that for any good notion of size , we’d have whenever a module is contained in a larger module (i.e. is a submodule of ). Dimension of vector spaces certainly has this property, but the minimum size of a generating set for modules does not.
Example. Take the polynomial ring in two variables . As an -module itself, can be generated by a single element, . But the submodule of generated by and (i.e. the elements of that are -linear combinations of and ) cannot be generated by fewer than two elements. So in this sense we have a smaller module with a bigger “size.”
In general, an -submodule of a ring is called an ideal of , and the ideal generated by a collection of elements is denoted . So denoting the minimal number of generators of an -module by , in the above example we have but .
If we modify the example so that is a polynomial ring in infinitely many variables, , we get an even more striking example: is still , but is now infinite.
Question: Do you need such a big ring to get an example of a finitely generated module with a submodule that isn’t finitely generated?
Answer: Yes! Rings with the property that their finitely generated modules only have finitely generated submodules are called Noetherian—so a polynomial ring in infinitely many variables is not Noetherian—and there are two main theorems that make it easy to prove rings are Noetherian:
- It’s sufficient to check only that all ideals of are finitely generated to prove that is Noetherian.
So since every ideal of can be generated by a single element—i.e. every ideal of is principal—we automatically know that every submodule of a finitely generated -module is also finitely generated.
The other theorem has to do with finitely generated algebras over a field, of which is one but isn’t. An algebra over a ring is just another ring equipped with a ring homomorphism ; a generating set for an algebra is a subset such that every element of can be written as a polynomial in the elements of with coefficients in . (In other words, the ring homomorphism from the polynomial ring to sending each to is surjective.) A finitely generated -algebra is just an algebra with a finite generating set, i.e. a quotient of the polynomial ring for some .
- Every finitely generated algebra over a field is Noetherian.
The proof goes by first checking that fields are Noetherian—they only have two ideals, and the whole field, which are both principal—then showing that if is a Noetherian ring then is Noetherian too (this is called the Hilbert Basis Theorem) and so is any quotient of . By induction, this gives Noetherianness of every finitely generated algebra over a field.
This means that any non-Noetherian algebra over a field must need infinitely many algebra generators. But in the Noetherian context, even though the number of generators of a module isn’t a good notion of size, we can still distinguish “small” modules (the finitely generated ones) from “large” ones (the rest), and a submodule of a small module is small.