Chapter 8 - Noetherian rings and modules

Summary

There is very little that one can say about a general ring and its modules. In practice an interesting structure theory will result either if the ring has a topology (which is compatible with its operations), or if it has finite dimension, or some generalization thereof. As an example of the former, we have the theory of C*-algebras. The latter class includes many important rings: algebras that are finite dimensional over a field, PI rings, artinian rings and noetherian rings. It is the last ones that we now study. In particular, we prove that the Weyl algebra is a noetherian ring.

NOETHERIAN MODULES.

In this book we shall be concerned almost exclusively with finitely generated modules. One easily checks that a homomorphic image of a finitely generated module is finitely generated. However a finitely generated module can have a submodule that is not itself finitely generated. An example is the polynomial ring in infinitely many variables K[x₁, x₂, …]. Taken as a module over itself this ring is a cyclic left module: it is generated by 1. However, the ideal generated by all the variables x₁, x₂, … cannot be finitely generated: every finite set of polynomials in K[x₁, x₂, …] uses up only finitely many of the variables.

We get around this problem with a definition. A left R-module is called noetherian if all its submodules are finitely generated. Examples are easy to come by: vector spaces over K are noetherian K-modules. Every ideal of the polynomial ring in one variable K[x] is a noetherian K[x]-module.

There are several equivalent ways to define noetherianness. We chose the most natural. Here are two more.