I'm reading Manin's article on formal groups and I have a problem with lemma 1.1. 
consider k a prefect ring of characteristic p and (A,m,k) a noetherian complete local ring of the same characteristic such that $X=spf\, A$ is a formal group. Manin wants to prove $X$ is a direct limit of finite group schemes over $k$.

define $m^{(p^n)}$ as the ideal generated by
$$\{x^{p^n}|x\in m\}$$
(Manin use the notation $m^{p^n}$ but I don't understand why $m^{p^n}$ generated by these elements.) he proves that $\frac{A}{m^{(p^n)}}$ are Hopf algebras and because $A$ is complete we have $A=\varprojlim \frac{A}{m^{(p^n)}}$ as rings.

I don't understand if $m^{(p^n)}$ is really equal to $m^{p^n}$ and if not why $\frac{A}{m^{(p^n)}}$ is finite over k and why A and $\varprojlim \frac{A}{m^{(p^n)}}$ are equal as topological rings?