Applications of: $R$ a subalgebra of $\mathrm{End}_k(V)$ s.t. $V$ is a simple $R$-module, then $R = \mathrm{End}_k(V)$. I am studying semisimple rings and modules. While doing that I encountered a theorem mentioned below. It states that:

Let $V$ be a finite-dimensional vector space over an algebraically closed field $k$. Let $R$ be a subalgebra of $\mathrm{End}_k(V)$. If $V$ is simple as an $R$-module, then $R = \mathrm{End}_k(V)$.

I understood this theorem very well and I know the proof too. My concern is how to use this theorem and I want to see some examples and consequence of this to understand it better. One consequence I remember is while proving that representation of a finite group over an algebraically closed field is contained in left-regular representation of $k[G]$. But I wanted to see some more direct applications.
Further I want to see a counterexample in the case where $V$ is not a finite-dimensional vector space over $k$.
 A: 
While doing that I encountered a theorem mentioned below.

This theorem is known as Burnside’s theorem on matrix algebras.

[…] I want to see some examples and consequence of this to understand it better.

The following is a common consequence of the theorem:
Corollary.
Let $$ be an algebraically closed field and let $A$ and $B$ be two $$-algebras.
Let $M$ and $N$ be finite-dimensional simple modules over $A$ and $B$ respectively.
Then $M ⊗_ N$ is simple as an $(A ⊗_ B)$-module.
Proof.
The module structures on $M$ and $N$ correspond to homomorphisms of $$-algebras
$$
  ρ_A \colon A \longrightarrow \mathrm{End}(M) \,,
  \quad
  ρ_B \colon B \longrightarrow \mathrm{End}(N) \,.
$$
The resulting $(A ⊗ B)$-module structure on $M ⊗ N$ corresponds to the homomorphism of $$-algebras
$$
  ρ_{A ⊗ B}
  \colon 
  A ⊗ B
  \xrightarrow{\enspace ρ_A ⊗ ρ_B \enspace}
  \mathrm{End}(M) ⊗ \mathrm{End}(N)
  \xrightarrow{\enspace f ⊗ g \mapsto f ⊗ g \enspace}
  \mathrm{End}(M ⊗ N) \,.
$$
It follows from Burnside’s theorem on matrix algebras that both $ρ_A$ and $ρ_B$ are surjective, whence the homomorphism $ρ_A ⊗ ρ_B$ is again surjective.
The homomorphism $\mathrm{End}(M) ⊗ \mathrm{End}(N) \to \mathrm{End}(M ⊗ N)$ is surjective because $M$ and $N$ are finite-dimensional.
It follows that $ρ_{A ⊗ B}$ is surjective.
Therefore, $M ⊗ N$ is simple as an $(A ⊗ B)$-module. $∎$

Further I want to see a counterexample in the case where $V$ is not a finite-dimensional vector space over $k$.

Let $V$ be any infinite-dimensional $$-vector space, where $$ is any field. Let $F$ be the two-sided ideal of $\mathrm{End}_(V)$ consisting of all endomorphisms of finite rank.
By adding $\mathrm{id}_V$ to $F$, we get the subalgebra $A ≔  ⋅ \mathrm{id}_V + F$ of $V$.
This is a proper subalgebra of $\mathrm{End}_(V)$.
The vector space $V$ is simple as an $F$-module (when we view $F$ as a non-unital $$-algebra), whence $V$ is also simple as an $A$-module.
To see that the above claims are true, let $(e_i)_{i ∈ I}$ be a basis of $V$.
For every two indices $i, j ∈ I$ there exists an endomorphism $e_{ij}$ of $V$  with $e_{ij}(e_k) = δ_{jk} e_i$.
These are endomorphism of rank $1$, whence all $e_{ij}$ are contained in $F$.
Let $W$ be a non-zero $F$-submodule of $V$, containing a non-zero element $w$.
For the linear combination $w = \sum_{i ∈ I} λ_i e_i$ some coefficient $λ_j$ is non-zero.
It follows that $e_j = e_{jj}(w) / λ_j$ is contained in $W$.
It further follows that $e_i = e_{ij}(e_j)$ is contained in $W$ for every index $i$.
This shows that $W = V$, which tells us that $V$ is simple as an $F$-module.
With respect to the basis $(e_i)_{i ∈ I}$ every endomorphism of $V$ can be represented as a column-finite $(I × I)$-matrix with coefficients in $$.
The ideal $F$ consists precisely of those endomorphisms whose matrix has only finitely many non-zero rows.
The identity endomorphism $\mathrm{id}_V$ is represented by the identity matrix.
We see that $A = F +  ⋅ \mathrm{id}_V$ is still a proper subspace of $\mathrm{End}(V)$.
