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?

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What is a module?

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