Commutativity of $\operatorname{End}_{R}(M)$ when $M$ is a semi-simple module Let $M$ be a semi-simple module over a unital ring $R$. I want to see if  $\operatorname{End}_{R}(M)$ is commutative only if $M$ is a direct sum of pair-wise non-isomorphic simple modules. 
 A: Suppose that $M \cong \bigoplus_{i \in I} M_i$ is a decomposition into simple submodules. Suppose that $M_{j_1} \cong M_{j_2}$ for some $j_1 \neq j_2$. Then $M_{j_1} \oplus M_{j_2} \subseteq M$ is a submodule, and we have an embedding
$$
 \Phi \colon \mathrm{End}_R(M_{j_1} \oplus M_{j_2})
 \hookrightarrow \mathrm{End}_R(M),
$$
where $\Phi(f)$ is given by
$$
 \Phi(f)\left( \sum_{i \in I} m_i \right)
= f(m_{j_1} + m_{j_2})
= f(m_{j_1}) + f(m_{j_2}),
$$
where $m_i \in M_i$. By Schur’s Lemma we have
$$
 \mathrm{End}_R(M_{j_1} \oplus M_{j_2}) \cong \mathrm{M}_2(D).
$$
for the skew field $D = \mathrm{End}_R(M_{j_1}) \cong \mathrm{End}_R(M_{j_2})$. Because $\mathrm{M}_2(D)$ is not commutative it follows that $\mathrm{End}_R(M)$ is not commutative.
So for $\mathrm{End}_R(M)$ to be commutative all simple summands of $M$ must be non-isomorphic.
PS: We can more generally look at a decomposition $M = \bigoplus_{i \in I} M_i^{(N_i)}$ with $M_i$ being simple and $M_i \ncong M_j$ for $i \neq j$ and $N_i$, $i \in I$ sets of the right cardinality. Using Schur’s Lemma we then have
$$
 \mathrm{End}_R(M)
 \cong \prod_{i \in I} \mathrm{End}_R\left( M_i^{(N_i)} \right)
 \cong \prod_{i \in I} \mathrm{M}_{N_i}(D_i)
$$
where $D_i =  \mathrm{End}_R(M_i)$ is a skew field and $\mathrm{M}_{N}(D)$ denotes the column finite $N \times N$ matrices with entries in $D$. So we can think of $\mathrm{End}_R(M)$ as the product of matrix rings over skew fields. For this to be commutative we need that all these matrix rings are actually of size $1 \times 1$, i.e $|N_i| = 1$ for all $i \in I$, and for all $i \in I$ the skew field $D_i$ must already be a field, i.e. be commutative.
(This idea of identifying $\mathrm{End}_R(M)$ with a product of matrix rings over skew fields is basically how the classification of semisimple rings (and semisimple algebras) works.)
