$\operatorname{End}(V)$ and $\operatorname{End}(V)\times\operatorname{End}(V)$ are isomorphic

Question

Let $R=\operatorname{End}(V)$ be the ring of all linear endomorphisms of an infinite dimension complex vector space $V$ with countable basis $\{e_{1},e_{2},...\}$ . Prove that $R$ and $R\times R$ are isomorphic as left $R$-modules.

Answer 1

Pick subspaces $S$ and $T$ of $V$ which are both of the same dimension as $V$ and such that $V=S\oplus T$. Decompose $End(V)$ with respect to that direct sum decomposition.