Dependencies among $\xi^{j}$

Question

Let $\xi\in GR(4^{m})$ and $\xi$ is of order $2^{m}-1$. If $\pm\xi^{j}\pm\xi^{k}$ is not invertible for $0\leq j<k\leq 2^{m}-1$, where $m\geq 2$, then why $\pm\xi^{j}\pm\xi^{k}=2\lambda$, where $\lambda\in GR(4^{m})$?

Answer 1

I guess $R = \operatorname{GR}(4^m)$ is the Galois ring of characteristic $4$ and order $4^m$. This ring is local with the maximal ideal $2R$. So for any $r\in R$, $r$ is invertible if and only if $r\notin 2R$.