$R/(x) \cong R/(x^2)$ implies $(x)=(x^2)$

Question

Let $x \in R$. I would like to know if it is true that if $$ R/(x) \cong R/(x^2) $$ as $R$-modules then $(x)=(x^2)$.

I was trying to play with the isomorphism and it's inverse but no luck so far. Any suggestions? Is it even true?

Answer 1

The $R$-module $R/(x)$ is annihilated by $x$, so also $R/(x^2)$ is; in particular $$ x(1+(x^2))=x+(x^2)=0+(x^2) $$ and so $x=x^2y$, for some $y$. (If $R$ is a domain, this implies $x=0$ or $x$ is invertible.)

But, yes: if $x=x^2y$, then $(x)\subseteq(x^2)$. The other inclusion is obvious.