Multi-stage Euclidean domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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Definition

Definition with symbols

Let $n$ be a positive integer.

An integral domain $R$ is said to be a $n$-stage Euclidean domain if there exists a function $N$ from the set of nonzero elements of $R$ to the set of nonnegative integers, such that for any $a$ and $b$ there exist $q_{i},r_{i}\in R$ for $1\leq i\leq n$such that:

$a=bq_{1}+r_{1}$

$b=q_{2}r_{1}+r_{2}$

and for $1\leq i\leq n-2$:

$r_{i}=q_{i+2}r_{i+1}+r_{i+1}$

such that either $r_{n}=0$ or $N(r_{n})<N(b)$.

Relation with other properties

Stronger properties

• Euclidean domain

Weaker properties

• Bezout domain