tough (*) question from a previous examination paper:
6.(*)
a) Let $R=\mathbf{C}[q]/q^{2}\mathbf{C}[q]$ and let $m,n \in \mathbf{N}$ where $R^{m},R^{n}$ are R-isomorphic. Show that m is equal to n.
b) Now let $R=\mathbf{Z}[5i]$ and $I=2R+(1+5i)R=2\mathbf{Z}+(1+5i)\mathbf{Z}$. Show that R and I are $\mathbf{Z}$-isomorphic but not R-isomorphic.
For a) I thought about showing that $\dim(R^{m})=\dim(R^{n})$, but I don't see how to do that (nor if it is a correct assumption). I lack an idea for b). Since solutions are not available, help is greatly appreciated.
edit: this theorem is probably relevant for the solution of the first one: $R^{m}$ isomorphic to a module M $\Leftrightarrow$ M has a basis with m elements.
Is there a way to use a) to solve b)?