Noetherian and Artinian Rings

Abstract

In this chapter we develop the theory of Noetherian and Artinian rings. In the first section, we will see that the Artin property, although in complete formal analogy to the Noether property, implies the Noether property and is, in fact, much more special (see Theorem 2.8). Both properties will also be considered for modules. In the second section, we concentrate on the Noether property. The most important results are Hilbert’s basis theorem (Corollary 2.13) and its consequences. Using the Noether property often yields elegant but noncon structive proofs. The most famous example is Hilbert’s proof [27] that rings of invariants of GLn and SLn are finitely generated, which for its nonconstruc tive nature drew sharp criticism from Gordan, the “king of invariant theory” at the time, who exclaimed, “Das ist Theologie und nicht Mathematik!”1