I had a post a few days back on why simple representations of algebras over a field {k} 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: {k} is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), {A} is a {k}-algebra, {Z} its center. We assume {Z} is a finitely generated ring over {k}, so in particular Noetherian: each ideal of {Z} is finitely generated.

Theorem 1 (Dixmier, Quillen) If {A} is a finite {Z}-module, then any simple {A}-module is a finite-dimensional {k}-vector space.