Let $_RM$ be a left $R$-module with $S=\text{End}_R(M)$, the ring of endomorphisms of the left $R$-module $M$ and $1$ be the identity endomorphism of $_RM$. It is well known that if $M=R$, then $R\cong S$. My questions are:
(1) Apart from the module $M=R$, which other types or classes of modules for which $S\cong R$ or $S=R$?
(2) For which classes of rings or types of ring for which every $R$-modules has $S\cong R$ or $S=R$?
(3) For which rings or modules every $\alpha\in S$ is only right multiplication, that is, $\alpha:M\to M$ is just $m\mapsto ma$ for some $a\in R$?