Submodule of cyclic module

Question

I am reading On $M$-Projective and $M$-Injective Modules by Azumayya (https://msp.org/pjm/1975/59-1/pjm-v59-n1-p02-s.pdf). There is a statement in the proof of Theorem 2.4: “every submodule of $Rm$ is of the form $Im$ with a suitable left ideal $I$”, with $M$ is an $R$-module and $m\in M$, $Rm$ is the cyclic submodule of $M$. I know $Im$ is a submodule of $Rm$, but is every submodule of $Rm$ in the form $Im$? Which means that every submodule of cyclic module is cyclic? Thank you for any help.

Answer 1

No: ${}_RR$ itself is a cyclic module, but of course it rarely has only cyclic submodules.

But consider the homomorphism $\phi: {}_RR \to {}_RRm$ given by $1\mapsto m$. By the first isomorphism theorem $R/L_m\cong Rm$. Each submodule in $Rm$ corresponds to a left ideal $I$ of $R$ containing $L_m$, whose image $\phi(I)=Im$.

You see, it simply does not follow that $Im$ has to be cyclic. It looks cyclic because everything has the form $im$, but do you think you can select a single $im\in Im$ such that $Rim=Im$?