I am starting to read about $p$-adic integers and one thing struck me. I have some related questions below. Can anybody help me out?
Can we have $n$-adic integers if we try to construct it in the same way? Like if we take $n=4$, then the number $10=2+4.2$, in $4$-adic integer becomes $22$?
If $n=pq$, does numbers represented in this $n$-adic form split into a number in $\mathbb Z_p\times\mathbb Z_q$, where $\mathbb Z_p,\mathbb Z_q$ are fields of $p$ and $q$-adic integers? The reason I am asking it is, clearly we can represent numbers in this form which I call $n$-adic integers, in which the number $a_0a_1...a_k$, $a_i\in \mathbb{Z}_n$, can be splitted into a tuple in $\mathbb Z_p\times\mathbb Z_q$, by mapping each $a_i$ to $a_i\;mod\;(p)$ and $a_i\;mod\;(q)$. Am I correct?