Rings and Modules

The course is a natural continuation of the course TMA4150 Algebra. Itgives a basis for further studies in algebra by discussing centralclasses of rings, including quotients of path algebras, and by givingan introduction to the theory of modules and representations ofquivers with relations. The course includes Artinian and Noetherianrings and modules and modules of finite length, where the latter isshown to be equivalent to modules which are both Artinian andNoetherian. Both structure theorems for simple and semisimple ringsand for modules over principal ideal domains are proved. As anapplication of the structure theorem of modules over principal idealdomains, proofs of rational canonical and Jordan canonical form of amatrix are given.

1. Knowledge: The student has knowledge of fundamental properties ofrings and modules, and examples of such. In particular, the student isfamiliar with the key properties and examples of noetherian, artinianand semisimple rings and modules, quotients of path algebras andrepresentations of quivers with relations, modules of finite length,and modules over principal ideal domains.

2. Skills: The student is able to decide whether a given ring ormodule, or a class of rings or modules, is noetherian, artinian orsemisimple, by applying the characterizations discussed in the course.The student is able to convert a module over a quotient of a pathalgebra to a representation of the corresponding quiver and theopposite direction. The student is able to calculate the rationalcanonical form and the Jordan canonical form of a matrix.

Teaching activities may vary depending on the lecturer. Students may answer either in Norwegian or English for assessments.