Let $m_1R,m_2R,m_3R,\ldots,m_nR$ be cyclic right $R$-module with unity. If every idempotent in the endomorphism ring $\text{End}_R(m_iR)$ is central, then are the idempotents of $\text{End}_R(M)$ where $M=m_1R+m_2R+m_3R+\ldots+m_nR$ central in $\text{End}_R(M)$? If not, then is there a counter example?
My attempt:
Since all the idempotents in $\text{End}_R(m_iR)$ are central, let $f,e^2=e\in \text{End}_R(M)$ and $m\in M$. Since $M=m_1R+m_2R+m_3R+\ldots+m_nR, f=\{f_i\}_{i=1}^{n}$ and $e=\{e_i\}_{i=1}^{n}$ for some $f_i,e^2_i=e_i\in \text{End}_R(m_iR)$.
An arbitrary element $m$ of $M$ is $\displaystyle{m=\sum_{i=1}^{n}x_i\in M}$ with $x_i=m_ir_i, m_i\in M,r_i\in R$ for each $i=1,\ldots,n$.
Then the centrality of $f$ is given by $\displaystyle{ef (m)=ef\left(\sum_{i=1}^{n}x_i\right)=\sum_{i=1}^{n}e_if_i(x_i)=\sum_{i=1}^{n}f_ie_i(x_i)=f e\left(\sum_{i=1}^{n}x_i\right)=f e(m)}$.
But I don't know whether such a definition of $f$ really makes sense.
Clearly not?
For example, for $R=M_2(\mathbb R)$, $R_R\cong S\oplus S$ where $S_R$ is a (the unique) simple (therefore cyclic) $R$ module. The $R$ endomorphism ring of $S$ is a field, so that $S$ is an abelian module, and clearly $R$ is not an abelian ring, so $S\oplus S$ is not an abelian module.