I was playing with some endomorphism rings and got curious whether a classification of all (not necessarily unitary) module $ M$ over a (not necessarily unital) ring $ R$ such that for every $ 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. 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?