Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Consider the ring $R=\mathbb{Z}_p[[T]]$. Let $f,g \in R$ and assume that $f=a_0+a_1T+...$ with $a_i \in p\mathbb{Z}_p$ for $0\le i \le n-1$, but $a_n \notin p\mathbb{Z}_p$. Then we may uniquely write $$g=qf+r$$ where $q \in R$ and $r \in \mathbb{Z}_p[T]$ is a polynomial of degree at most $n-1$
The proof of this result may be found in Washington's "Introduction to Cyclotomic Fields". I am interested in a division algorithm for the ring $S=\mathbb{Z}_p[[T_1, T_2,...,T_m]]$. Is there an appropriate formulation of something like the above result for this ring $S$?
