Consider the group-ring $\mathbb{Z}_p[\mathbb{Z}/p^{n}\mathbb{Z}]$ with the product topology, and the quotient ring $\mathbb{Z}_p[T]/((1+T)^{p^n}-1)$ with the quotient topology, ($\mathbb{Z}_p[T]$ has the $(p,T)$-adic topology).

If $\gamma$ is a generator of $\mathbb{Z}/p^{n}\mathbb{Z}$, then clearly the map: $$\gamma\mapsto 1+T$$ is a ring isomorphism. My question is the following: ¿that map is also a homeomorphism? I'm not even sure if $\mathbb{Z}_p[T]/((1+T)^{p^n}-1)$ is Hausdorff. (Maybe i'm giving $\mathbb{Z}_p[T]/((1+T)^{p^n}-1)$ the wrong topology).

Any hint would be greatly appreciated.


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