# 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?

The most familiar example of a module is probably a vector space like $\mathbb{R}^n$, in which you can add and subtract vectors, or multiply a vector by a real number (a scalar). More generally, the abstract axioms for a vector space $V$ over a field $K$ look like this: Continue reading