# Dimension of $End(V)$ with $V$ countable dimension irreducible module over a complex algebra

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.

• I made some mistakes I've just edited; however I mean irreducible module, that is a module with no bilateral submodules except himself and $(0)$. – Giuseppe Bargagnati Dec 22 '17 at 14:35
• It is much clearer now, thanks for the edits. – rschwieb Dec 22 '17 at 15:12
• 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. – rschwieb Dec 22 '17 at 15:26
• What is “dimension” “as $A$-module”? Also, what is “countable” - infinite or not necessarily? – Dap Dec 29 '17 at 7:37
• How is $End(V)$ an $A$-module? – Dap Dec 30 '17 at 14:09

## 1 Answer

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 $$\{z \in \mathbb{C}: a - z \text{ is not invertible}\}$$.