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.

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Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago.  It was a useful bit of algebra for me to think about.

Theorem 1 Let ${A}$ be an associative algebra with identity over an algebraically closed field ${k}$; suppose the center ${Z \subset A}$ is a finitely generated ring over ${k}$, and ${A}$ is a finitely generated ${Z}$-module. Then: all simple ${A}$-modules are finite-dimensional ${k}$-vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.