I had a post a few days back on why simple representations of algebras over a field which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.
So, let’s review the notation: is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post),
is a
-algebra,
its center. We assume
is a finitely generated ring over
, so in particular Noetherian: each ideal of
is finitely generated.
Theorem 1 (Dixmier, Quillen) If
is a finite
-module, then any simple
-module is a finite-dimensional
-vector space.