Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Question

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)

Answer 1

I'm not familiar with topological generators in a ring setting. I assume one wants the closure of the subring generated by an element to be the whole topological ring. We should have

$$\begin{array}{ll} \overline{\langle 1+T\rangle} & =\overline{(1+T)\Bbb Z[1+T]} \\ & =\overline{(1+T)\Bbb Z[T]} \\ & =(1+T)\overline{\Bbb Z[T]} \\ & =(1+T)\Bbb Z_p[[T]] \\ & =\Bbb Z_p[[T]] \end{array} $$

because $1+T\in\Bbb Z_p[[T]]^\times$ (whose inverse is provided by the geometric sum formula).