I am currently preparing a second edition of my introductory book on module theory, and for editorial reasons, I need to add 21 pages (the format of the pages is something like the new editions of Bourbaki), including few exercises.
The book aims to an audience of 4th year student, having a background in basic algebra (linear algebra over a field, a bit of group theory, basic ring theory).
For the moment, the book covers the following topics: modules, submodules, exact sequences, quotient modules, cyclic modules, definitions and basic properties of noetherian, free, projective, stably free modules, localisation, local characterization of projective modules, tensor product over a commutative ring, algebras and graded tensor product of algebras, tensor algebra, exterior algebra, exterior powers, fg modules over a PID, fg modules over a Dedekind domain, and Quillen Suslin Theorem (every finitely generated module over a polynomial ring is free).
The purpose of my book is to be a first course in module theory, so the last two topics are really the maximal level of difficulty I want to reach.
The first idea coming to my mind is to add a new chapter, but i'm lacking of ideas considering the choice of the topic. I'm pretty sure I don't want to talk about representation theory, since 21 pages is too short to present something convincing.
I thought about adding a chapter on skew fields (since most of the theorems involve module theory and tensor products), but without any real application of skew fields or Brauer groups, it might feel that this chapter would pop out of nowhere (plus the fact that I left the proofs of Wedderburn theorem and Skolem Noether as exercices in the chapter on algebras).
Question. Do you have any idea of a suitable topic for a new chapter ?