Category:Definitions/Euclidean Valuations

This category contains definitions related to Euclidean Valuations.
Related results can be found in Category:Euclidean Valuations.

Let $\struct {D, +, \circ}$ be an integral domain with zero $0_D$.

Let there exist a mapping $\nu: D \setminus \set {0_D} \to \N$ such that for all $a \in D, b \in D_{\ne 0_D}$:

$(1)$	$:$			$\ds \exists q, r \in D: \map \nu r < \map \nu b \text { or } r = 0_D:$	$\ds a = q \circ b + r $
$(2)$	$:$				$\ds \map \nu a \le \map \nu {a \circ b} $

Then $\nu$ is a Euclidean valuation on $D$.

Pages in category "Definitions/Euclidean Valuations"

The following 5 pages are in this category, out of 5 total.

\((1)\)	$:$			\(\ds \exists q, r \in D: \map \nu r < \map \nu b \text { or } r = 0_D:\)	\(\ds a = q \circ b + r \)
\((2)\)	$:$				\(\ds \map \nu a \le \map \nu {a \circ b} \)