# Isomorphism of R-modules

Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?

#### Solutions Collecting From Web of "Isomorphism of R-modules"

Let $\mathrm{CFM}_\mathbb{N}(R)$ denote the ring of “column finite matrices”, where $R$ is some ring.. Then, one can show that $\mathrm{CFM}_\mathbb{N}(R)\to\mathrm{CFM}_\mathbb{N}(R)^2$ defined by

$$M\mapsto (\text{odd indexed columns of }M,\text{even indexed columns of }M)$$

is an isomorphism of $\mathrm{CFM}_\mathbb{N}(R)$-modules. It clearly then follows that $\mathrm{CFM}_\mathbb{N}(R)^m\cong\mathrm{CFM}_\mathbb{N}(R)^n$ for all $n$ and $m$.

I believe this example is in Dummit and Foote.