Let \(R\) be a ring and \(M\) an \(R\)-module. \(M\) is cyclic if there exists \(m\in M\) with \(M = Rm\coloneqq\theset{rm \st r\in R}\).
An \(R\)-module \(M\) is cyclic if and only if \(M \cong R/I\) for some left ideal \(I\) of \(R\).
If \(M=Rm\), the \(R\)-linear map \(\phi\colon R\to M\), \(r\mapsto rm\), is surjective, and its kernel \(I\coloneqq\theset{r\in R \st rm=0}\) is a left ideal, so \(M\cong R/I\) by the first isomorphism theorem. Conversely, \(R/I=R\cdot(1+I)\) is cyclic, and an isomorphism \(R/I\to M\) sends the generator \(1+I\) to a generator of \(M\).
When is an $R\dash$module cyclic, and what does it look like as a quotient of $R$?