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Let $A$ be a $\mathbb{C}$-algebra and $V$ be an irreducible $A$-module with countable dimension. What is the dimension of $End(V)$ as $A$-module? Note that every endomorphism must be injective and surjective, because of the irreducibility of $V$. My claim (or at least my hope) is that $End(V)$ has countable dimension, but I can't see a proof of this fact.

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  • $\begingroup$ I made some mistakes I've just edited; however I mean irreducible module, that is a module with no bilateral submodules except himself and $(0)$. $\endgroup$ – Giuseppe Bargagnati Dec 22 '17 at 14:35
  • $\begingroup$ It is much clearer now, thanks for the edits. $\endgroup$ – rschwieb Dec 22 '17 at 15:12
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    $\begingroup$ Without loss of generality you can assume $V$ is a faithful $A$ module and think about right primitive rings and apply the Jacobson density theorem. $\endgroup$ – rschwieb Dec 22 '17 at 15:26
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    $\begingroup$ What is “dimension” “as $A$-module”? Also, what is “countable” - infinite or not necessarily? $\endgroup$ – Dap Dec 29 '17 at 7:37
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    $\begingroup$ How is $End(V)$ an $A$-module? $\endgroup$ – Dap Dec 30 '17 at 14:09
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I have tried to interpret the question as asking about dimension over $\mathbb{C}$ , as it is not clear what $A$-module structure $End_A(V)$ has.

If $V$ is a left $A$-module which is irreducible in the sense that it has no proper nontrivial left $A$-modules, then $\dim_{\mathbb{C}} End_A(V) \leq \dim_{\mathbb C} V$. For if $v \in V \setminus \{0\}$, then $V = Av$, and thus the $\mathbb{C}$-linear map \begin{align*} End_A(V) &\to V \\ \varphi &\mapsto \varphi(v) \end{align*} is injective.

Schur's Lemma tells us that every nonzero endomorphism in $End_A(V)$ is in fact invertible, that is, that $End_A(V)$ is a division algebra over $\mathbb{C}$ of countable dimension. While it is well known that the only finite-dimensional division algebra over $\mathbb{C}$ is $\mathbb{C}$ itself, the same result is true of division algebras $D$ of countable dimension, as can be seen by examining for $a \in D \setminus \mathbb{C}$ the set $\{z \in \mathbb{C}: a - z \text{ is not invertible}\}$.

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