Let $R$ be a commutative ring with unity possessing an element $r$ in the singular ideal $Z(R)=$ the set of elements whose annihilators are essential in the module $R_R$, and let $M$ be a faithful $R$-module. Could it be possible to define an $R$-module monomorphism $f:M\longrightarrow R$?
If the answer is in affirmative, one could prove, first, that $f^{-1}(\mathrm{Ann}_R(r))=\mathrm{Ann}_M(r)$, where $\mathrm{Ann}$ stands for annihilator, whence $\mathrm{Ann}_M(r)$ is essential in $M$. The r.h.s. is easily seen to be a subset of the l.h.s.; for the reverse, if $\mu \in f^{-1}(\mathrm{Ann}_R(r)$ then $f(r\mu)=0$ and so $r\mu=0$.