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Representation of an ideal class in $\widehat{\mathbb{Z}}$

Let $n$ be an integer. Let $I(n)$ be the ideal of $R=\mathbb{Z}[\zeta_8]$ generated by $1-\zeta_8^n$. Let $J(n)$ be the maximal ideal of $R$.
Then
$I(n) = (1-\zeta_8^n)R$ where $R=\mathbb{Z}[\zeta_8]$ and $\mathbb{Z}[\zeta_8]=\{a+b\zeta_8:a,b\in \mathbb{Z}\}$.
Consider the ring $\mathbb{Z}/J(n)\mathbb{Z}$. It is of order $2$ and it is isomorphic to $\mathbb{Z}[\zeta_4]$. In this ring, we can represent the ideal $I(n)$ as $(1-\zeta_4^n)R$.
Question :
Since $\mathbb{Z}/J(n)\mathbb{Z}\simeq \mathbb{Z}[\zeta_4]$ and since $a+b\zeta_4=a+b\zeta_4^3$, can we represent $I(n)$ as $(1-\zeta_4^n)R$ in $\mathbb{Z}/J(n)\mathbb{Z}$?
If the answer is yes, I’d also like to know the proof. If it is no, then is there a way to understand why the answer is no? In particular, the representation of $I(n)$ is important for me.
Edit:
With $\zeta_n=e^{2\pi i/n}$, $I(n)$ is the ideal of $R$ generated by $1-\zeta_8^n$. So \$I