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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.

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    @QiL: I can't thank you enough! I really really wanted to solve this question, I even broke down at one point over it! Maybe if you post anything as an answer, I can select it as best and give you some more points? – 2011-11-16

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As required by the OP, here is the hint for solve the problem: show that $N(a)=p^{nāˆ’1}$ if and only if $a$ is $p^{nāˆ’1}$ times a unit of $R$.

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    thank you again! It seems like the right answer, a fellow student also proved it that way :) – 2011-11-16