The euclidean-domain tag has no usage guidance.

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### Intersection of powers of prime ideals

Let $R$ be a Noetherian ring. Let $(x)$ be a prime ideal such that $\bigcap_n (x)^n=0$. Then $R$ is a domain.
Is this a known result? I heard its known as the Davis lemma. Can anyone give a reference?...

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### How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient
algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that
$n = \sum_{i=1}^k a_i m_i$?
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### Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...

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### Reference request: number theory of Z[1/p]

Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in the denominator)?
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### Must a ring which admits a Euclidean quadratic form be Euclidean?

The question is in the title, but employs some private terminology, so I had better explain.
Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R \setminus \{0\}$. ...

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### Why do we care whether a PID admits some crazy Euclidean norm?

An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$,...