$\newcommand{\End}{\operatorname{End}}\newcommand{\Hom}{\operatorname{Hom}}$ On pg. 647 of Lang's Algebra, Lang proves the Jacobson density theorem by doing some stuff with $\End_{\End_R(V^n)}(V^n)$ and relating $\End_R(V^n)$ to matrices with coefficients in $R$ (excerpted below)
Lang presents as a corollary Weddurburn's theorem, which is a "double centralizer" type theorem. Googling "double centralizer theorem rings" produces Double centralizer theorem for semisimple rings and also Double Centralizer Property in Modules over Semisimple Rings which presents a theorem very very similar to Lang's "Weddurburn's theorem" (the only difference is that the MSE questions assumes the ring is semisimple, whereas Lang merely assumes the module is semisimple).
There's also this MSE question Jacobson's Density Theorem for Semisimple Algebras, which links this set of notes: https://math.mit.edu/~etingof/replect.pdf. This set of notes focuses on algebras $A$ over a field $k$ (perhaps required to be algebraically closed, though I'm not sure exactly where this is used). For a $k$-VS (vector space) $V$, a (left) $A$-module structure is equivalent to specifying a $k$-algebra homomorphism $\rho: A \to \End_k(V)$, i.e. a representation of the $k$-algebra $A$. I will copy over some relevant results:
Proposition 2.2. Let $V_{i}, 1 \leq i \leq m$ be irreducible finite dimensional pairwise nonisomorphic representations of $A$, and $W$ be a subrepresentation of $V=\oplus_{i=1}^{m} n_{i} V_{i}$. Then $W$ is isomorphic to $\oplus_{i=1}^{m} r_{i} V_{i}, r_{i} \leq n_{i}$, and the inclusion $\phi: W \rightarrow V$ is a direct sum of inclusions $\phi_{i}: r_{i} V_{i} \rightarrow n_{i} V_{i}$ given by multiplication of a row vector of elements of $V_{i}$ (of length $r_{i}$ ) by a certain $r_{i}$-by-n $n_{i}$ matrix $X_{i}$ with linearly independent rows: $\phi\left(v_{1}, \ldots, v_{r_{i}}\right)=\left(v_{1}, \ldots, v_{r_{i}}\right) X_{i}$.
Let A be an algebra over an algebraically closed field k.
Corollary 2.4. Let $V$ be an irreducible finite dimensional representation of $A$, and $v_{1}, \ldots, v_{n} \in V$ be any $k$-linearly independent vectors. Then for any $w_{1}, \ldots, w_{n} \in V$ there exists an element $a\in A$ such that $a v_{i}=w_{i}$
Theorem 2.5. (the Density Theorem). (i) Let $V$ be an irreducible finite dimensional representation of $A$. Then the map $\rho: A \rightarrow \End_k(V)$ is surjective. (ii) Let $V=V_{1} \oplus \ldots \oplus V_{r}$, where $V_{i}$ are irreducible pairwise nonisomorphic finite dimensional representations of $A$. Then the $\operatorname{map} \bigoplus_{i=1}^{r} \rho_{i}: A \rightarrow \bigoplus_{i=1}^{r} \operatorname{End}\left(V_{i}\right)$ is surjective.
My questions are as follows:
How exactly does the Jacobson density theorem relate to theorems like Artin-Wedderburn? Can it be proven from A-W?
- It looks like Prop. 22 I copied above (disregarding the statement about matrices for now) should follow easily from Artin-Weddurburn, or at least from the same tools as A-W... e.g. we would know something very explicit like $r_i = \dim_{D_i}\Hom_A(V_i,W)$ for division rings $D_i := \End_A(V_i)$. From this Cor. 2.4 follows (I wonder if there is a more elegant proof without the matrices, but whatever --- also I don't know where $k$ being algebraically closed is used!), and from that the density theorem (Thm. 2.5) follows easily.
- Also in the case that $A$ is a semisimple ring A-W gives exactly $A \simeq M_{n_1}(D_1) \times \ldots \times M_{n_m}(D_m)$ for $n_i = \dim_{D_i}\Hom_A(V_i,A)$ (isomorphic as $k$-algebras and as (left) $A$-modules). IF this isomorphism is via $\rho_1 \oplus \ldots \oplus \rho_m$ (for the representation $\rho_i$ corresponding to irreducible rep. $V_i$, i.e. $\rho_i(a)$ is a $k$-linear self-map of the $k$-VS $V_i \simeq D_i^{n_i}$, which can be identified as a matrix in $M_{n_i}(D_i)$), then of course we have surjectivity. Is this proof sketch right, i.e. in particular the density theorem (Thm. 2.5) follows from A-W?
Is there some "unifying perspective" on Lang's Jacobson density theorem, the double centralizer theorem for semisimple rings in the MSE post I linked, and the double centralizer theorem for central simple algebras?