query: Definition of a Euclidean Domain: how to show that the condition $f(ab)\ge f(a)$ is actually not necessary?
positive_passages.docid: 17161
positive_passages.text: Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?
negative_passages.docid: 154845
negative_passages.text: Irreducible elements in  $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?
negative_passages.docid: 123024
negative_passages.text: A Noetherian integral domain is a UFD iff $(f):(g)$ is principal
negative_passages.docid: 2129
negative_passages.text: Showing that the derivative operator is not bounded on $L^2 (\mathbb{R})$
negative_passages.docid: 111930
negative_passages.text: Domain of a Standard Parabola
negative_passages.docid: 68746
negative_passages.text: What is $f(f^{-1}(A))$?
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 126503
negative_passages.text: Showing that an integral domain is a PID if it satisfies two conditions
negative_passages.docid: 4507
negative_passages.text: Localization of a ring which is not a domain
negative_passages.docid: 116453
negative_passages.text: Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
negative_passages.docid: 54321
negative_passages.text: Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
negative_passages.docid: 103550
negative_passages.text: Multiplicative Euclidean Function for an Euclidean Domain
negative_passages.docid: 125043
negative_passages.text: Show that F(A−B) is not necessarily a subset of F(A)−F(B).
negative_passages.docid: 23721
negative_passages.text: Prove that the Gaussian Integer's ring is a Euclidean domain
negative_passages.docid: 47975
negative_passages.text: Closed cone in the euclidean space $\mathbb{R}^n$
negative_passages.docid: 12272
negative_passages.text: Why $a$ is not an essential singularity and how $f$ can be extended
negative_passages.docid: 147722
negative_passages.text: Is there a nice example of integral domain $R$ and $s,t \in R$, with $Rs \cap Rt$ not principal but lcm($s,t$) exists?