Suppose $k$ be a field of characteristic $0$, and $R$ be an unital $k$-algebra. Then, does it hold $$ R \otimes_k \mathrm{End}_R(W) \cong \mathrm{End}_k(W)? $$ for an $R$-module $W$?
This is an attempt to understand the theory of Clifford modules, since there is no proof of this on any literatures. I'm especially interested in the case, $$ R = \mathbb{C}l(V)\\ k = \mathbb{C}\ $$
Might be related : This post, but I couldn't see the direct relation.
The identity in the title may be too general, but I can show the case you are interested in:
Let $V$ be an even-dimensional Euclidean vector space. The complexification of $C(V)$ (a complex algebra) has the special property that it is isomorphic to $\mathrm{End}(S)$ for some complex vector space $S$, see proposition 3.19 in [1]. So we want to show that $$\mathrm{End}(E)\cong \mathrm{End}(S)\otimes\mathrm{End}_{\mathrm{End}(S)}(E)$$ for an $\mathrm{End}(S)$-module $E$. The key observation is that such an $E$ is isomorphic to the $\mathrm{End}(S)$-module $S\otimes W$ for some complex vector space $W$ (see this answer). Hence $$\mathrm{End}(E)\cong \mathrm{End}(S\otimes W)\cong \mathrm{End}(S)\otimes\mathrm{End}( W)$$ and now the desired result follows from the fact that $$\mathrm{End}( W)\cong \mathrm{End}_{\mathrm{End}(S)}(S\otimes W)\cong \mathrm{End}_{\mathrm{End}(S)}(E).$$
[1] Heat Kernels and Dirac Operators