Let $R\subset \mathbb{Z}[[T]]$ be a subring of the the ring of power series over the integers. We define $R$ with the following property: $$ \sum a_iT^i\in R$$ if and only if there exist a $L>0$ with $\mid a_i\mid\leq L$ for every coefficent $a_i$.

I choose this ring because for $p,q\in \mathbb{Z}$ the ideal generated by $pT-q$ does not contain the unity.

Let $q>p>0$ be prime integers. Let $I_{\infty}$ be the ideal of power series that vanish in $p/q$ with the real topology. Let $I_{p}$ be the ideal of power series that vanish in $p/q$ with the p-adic topology. And let $I$ be the ideal generated by $pT-q$.

I have two question about this ring:

The first question is if $I$ is a prime ideal.

The second question follows as a consequence of the first question. My question is if $I=I_{\infty}= I_p$.

I do this question because I want to understand if there is a relationship beetwen the convergence over the p-adic topology and the real topology.