3
$\begingroup$

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 $r \geq 2$.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.