Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $r \geq 2$.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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