By the definition from Barnard (1980), a multiplication module is an $R$-module $M$ in which for all submodules $N$, there exists an ideal $I$ of $R$ such that $N=IM$. Note that a cyclic $R$-module (a module generated by one element), say $\langle m\rangle$ is a multiplication module since all of its submodules, say $N$, can be written as $Im=I(Rm)$ where $I=\langle r_i\ |\ r_im\in N\rangle$.
My question is: is there any example for a non-cyclic multiplication module? I've tried some examples such as $M_{2\times1}(\mathbb Z)$ over $M_{2\times2}(\mathbb Z)$, since I thought it is non-cyclic over $\mathbb Z$, but it turns out that it is cyclic over $M_{2\times2}(\mathbb Z)$. Is there any basic example of a non-cylic multiplication module? Thanks in advance!
Two easy examples:
(1) An invertible ideal is a multiplication module. Indeed, if $I$ is invertible and $J \subseteq I$ then $K = I^{-1}J \subseteq R$ is an ideal and $J = IK$. This points to any Dedekind domain which is not a PID as a source of multiplicative ideals that are not principal.
(2) Every ideal generated by idempotents is a multiplication module. This is also easy to check. So for example, in any infinite product of rings $\prod R_\alpha$, the ideal generated by the standard basis vectors is a multiplicative module which is not principal.