so I am currently learning about modules, I'm pretty new to them, but i have some experience with rings and linear algebra and stuff. I got the following problem:

Let $R$ be the ring $\mathbb{Z}[\sqrt{-5}]$ and $I$ be the ideal $\langle 2,1+ \sqrt{-5} \rangle$ in $R$. Show that $I^2$ is $R$-module isomorphic to $R^2$. Is there some $n \in \mathbb{N}_0$ such that $I$ is $R$-module isomorphic to $R^n$?

I managed to show that $I^2$ is the same as $\langle 2 \rangle$ just by multiplying things out and basically showing $I^2 \subseteq \langle 2 \rangle$ and $\langle 2 \rangle \subseteq I^2$. Though I don't really know how to progress from here or whether that is even useful. I need to find an isomorphism, but I'm not sure how to. multiplying by $2$ for example won't work, as the only units in $\mathbb{Z}[\sqrt{-5}]$ are $\pm 1$. Any help would be really appreciated!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.