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Consider the ring $\mathbb{Z}_p[[X]] \otimes_\mathbb{Z} \mathbb{Q}_p$ and the ideal generated by $(X-p)^r$ (for some integer $r$).

Is the following true : for all integer $r$, the ring $$ \frac{\mathbb{Z}_p[[X]] \otimes_\mathbb{Z} \mathbb{Q}_p}{(X-p)^r} $$ is principal ?

I can see that for $r=1$, the ring is isomorphic to $\mathbb{Q_p}$ (so it is principal), but I don't know when $r \geq 2$.

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1 Answer 1

up vote 4 down vote accepted

First of all, the change of variables $T = X - p$ yields an isomorphism $$\mathbb Z_p[[X]] \cong \mathbb Z_p[[T]].$$ (The point here is that $p$ lies in the maximal ideal of $\mathbb Z_p$.)

So your ring is isomorphic to $\mathbb Q_p[T]/(T^r),$ which you can easily verify is principal.

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