Noetherian rings and modules

Abstract

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 [ x 1 , x 2 , …]. 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 x 1 , x 2 , … cannot be finitely generated: every finite set of polynomials in K [ x 1 , x 2 , …] 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.