Let $R$ be a commutative ring with identity, and $I$ an ideal of $R$. Is it true that we have an $R$-module isomorphism
$$I\cong R/ann_RI,$$
where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the annihilator of $I$ in $R$?
If $I=(a)$ is a principal ideal, then $\varphi:R\rightarrow I$ defined by $\varphi (x)=ax$ is an $R$-module homomorphism with $\ker(\varphi)=ann_RI$ and $Im(\varphi)=I$. Therefore, by the first isomorphism theorem for modules we get the result. But if the ideal $I$ is not principal?
Best Answer
For any ring $R$, and any ideal $J$, $R/J$, as an $R$-module, is cyclic, meaning it is generated by a single element, namely the coset of $1$, $1+J$. But certainly there are $R$-modules that aren't cyclic. An ideal is cyclic if and only if it is principal, so, excluding rings where every ideal is principal, you have many examples of ideals for which this cannot possibly be true.