The ascending chain condition and existence of prime factorisations
In this video, we introduce the important algebraic notion of the ascending chain condition (ACC). Modules and rings satisfying this condition are said to be noetherian. We first motivate this concept by seeing how our usual argument for showing existence of prime factorisations for integers can be mimicked in the setting for prime factorisation for ideals in a Dedekind domain, as long as ACC holds. We give many basic properties of noetherian modules which show fairly precisely how ACC is a finiteness condition that is stronger than being finitely generated, but more stable since submodules and quotients of noetherian modules are noetherian. We see that Dedekind domains are noetherian thus establishing prime factorisation in this setting. We finish by giving many examples of noetherian rings, including Hilbert's basis theorem, a milestone in the subject which says that polynomial rings are noetherian if their coefficient rings are. This shows that the class of noetherian rings is large, rich and includes many examples which are of importance in mathematics.