Multi-stage Euclidean domain

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 $$1 \le i \le n$$such that:

$$a = bq_1 + r_1$$

$$b = q_2r_1 + r_2$$

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

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

such that either $$r_n = 0$$ or $$N(r_n) < N(b)$$.

Stronger properties

 * Euclidean domain

Weaker properties

 * Bezout domain