# What is a Noetherian ring?

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 $R$-module $M$, though, like $\{1, 1/2, 1/3, \dots\}$ for the $\mathbb{Z}$-module $\mathbb{Q}$: a subset of $M$ for which every element can be written as an $R$-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 $\mu$, we’d have $\mu(M)\leq\mu(N)$ whenever a module $M$ is contained in a larger module $N$ (i.e. $M$ is a submodule of $N$). 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 $R = \mathbb{R}[x,y]$. As an $R$-module itself, $R$ can be generated by a single element, $1$. But the submodule of $R$ generated by $x$ and $y$ (i.e. the elements of $R$ that are $R$-linear combinations of $x$ and $y$) cannot be generated by fewer than two elements. So in this sense we have a smaller module with a bigger “size.”

In general, an $R$-submodule of a ring $R$ is called an ideal of $R$, and the ideal generated by a collection of elements $\{a_1,a_2,\dots\}$ is denoted $(a_1,a_2,\dots)$. So denoting the minimal number of generators of an $R$-module $M$ by $\mu_R(M)$, in the above example we have $\mu_R(R) = 1$ but $\mu_R((x,y)) = 2$.

If we modify the example so that $R$ is a polynomial ring in infinitely many variables, $R=\mathbb{R}[x_1,x_2,\dots]$, we get an even more striking example: $\mu_R(R)$ is still $1$, but $\mu_R((x_1,x_2,\dots))$ 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:

1. It’s sufficient to check only that all ideals of $R$ are finitely generated to prove that $R$ is Noetherian.

So since every ideal of $\mathbb{Z}$ can be generated by a single element—i.e. every ideal of $\mathbb{Z}$ is principal—we automatically know that every submodule of a finitely generated $\mathbb{Z}$-module is also finitely generated.

The other theorem has to do with finitely generated algebras over a field, of which $\mathbb{R}[x,y]$ is one but $\mathbb{R}[x_1,x_2,\dots]$ isn’t. An algebra over a ring $R$ is just another ring $A$ equipped with a ring homomorphism $R\to A$; a generating set for an algebra is a subset $\{a_i:i\in I\}\subseteq A$ such that every element of $A$ can be written as a polynomial in the elements of $a_i$ with coefficients in $R$. (In other words, the ring homomorphism from the polynomial ring $R[x_i:i\in I]$ to $A$ sending each $x_i$ to $a_i$ is surjective.) A finitely generated $R$-algebra is just an algebra with a finite generating set, i.e. a quotient of the polynomial ring $\mathbb{R}[x_1,\dots,x_n]$ for some $n$.

1. Every finitely generated algebra over a field is Noetherian.

The proof goes by first checking that fields are Noetherian—they only have two ideals, $\{0\}$ and the whole field, which are both principal—then showing that if $R$ is a Noetherian ring then $R[x]$ is Noetherian too (this is called the Hilbert Basis Theorem) and so is any quotient of $R$. 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.