Let $ p \in \mathbb{Z}$ be a prime, and define
$$R:=\lbrace a=(a_1,a_2,a_3,\ldots) | a_k \in(\mathbb{Z}/p^k\mathbb{Z})\text{ and }a_{k+1}\equiv a_k \pmod {p^k}\text{ for all }k \in \mathbb{N} \rbrace$$
I have proved that R is a ring, with multiplication and addition defined component wise. The Norm for a= ($a_1,a_2,\ldots)\in R$ with $a\neq0$ is defined as $N(a)=p^{n-1}$ where $n$ is the smallest value of $k$ such that $a_k\neq0$ i.e. $a_n$ is the first non-zero term in the sequence $a$.
I need to prove that, if R is endowed with the map $N: R^*\to\mathbb{N}$, then R is an Euclidean domain.
Best Answer
As required by the OP, here is the hint for solve the problem: show that $N(a)=p^{n−1}$ if and only if $a$ is $p^{n−1}$ times a unit of $R$.