Commutative algebra 24 Artinian modules

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We define Artinian rings and modules, and give several examples of them. We then study finite length modules, show that they are the same as modules that are both Artinian and Noetherian, and conclude by showing that their length is well defined. Reading: Section 2.4 Exercises: 2.23 Classify the modules of finite length over k[x]. Find an Artinian module over this ring that is not Noetherian.