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I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ such that for every right $R$-module homomorphism $\varphi:M\to M$, there exists $r\in R$ such that $$\varphi(x)=r\cdot x$$ for all $x\in M$. I know that it means $$\text{End}_R(M)\cong R/\text{Ann}_R(M)$$ but I can't make any interesting conclusion from this. Here $$\text{Ann}_R(M):=\Big\{r\in R\,\Big|\,r\cdot x=0\text{ for every }x\in M\Big\}$$ is the left annihilator of $M$.

In the case that $R$ is a division ring and $M$ is an $R$-vector space, then it is obvious that $M$ must be at most one-dimensional. Could you please give me some references if there are any studies regarding this question?

Edit: I made an error, which has been fixed. I originally said that $M$ was a left $R$-module, which would make no sense if $R$ is non-commutative.

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    $\begingroup$ In the context of commutative domains, ideals whose endomorphism rings are isomorphic to the domain are called Archimedean (E. Matlis, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959) 57–69.) $\endgroup$
    – Luc Guyot
    May 7, 2019 at 15:42

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