From Wikipedia:
Let $R$ be an integral domain. A Euclidean function on $R$ is a function
$$f:R\setminus \{0\}\rightarrow\mathbb{Z}^+$$
satisfying the following fundamental division-with-remainder property:If $a$ and $b$ are in $R$ and $b$ is nonzero, then there are $q$ and $r$ in $R$ such that $a = bq + r$ and either $r = 0$ or $f(r) < f(b)$.
What is the motivation for the requirement $f(r)<f(b)$? (What is it good for?)
I immediately think of the Euclidean algorithm, which ensures us that the process of finding the $\gcd(a,b)$ must end. Is there a similar reason for $f(r)<f(b)$, to ensure that "some process" ends? I also thought about the proof that every Euclidean domain is a PID, where the proof chooses $b$ minimum so that $f(r)\not <f(b)\Rightarrow r=0$.
Best Answer
This is exactly the generalization of the standard division algorithm for integers. Suppose we have $n, m \in \mathbb{Z}$. We learn early on that there exists a unique $q \in \mathbb{Z}$ (called the quotient) and a unique $r \in \mathbb{Z}$ such that $$ n = mq + r, \ \ \ \ 0 \leq r < m. $$ The point is that we have a remainder $r$ that is less than the number $m$ which we divided $n$ by.
To generalize this to an arbitrary ring, where there is no a priori ordering, we need to find some condition to replace $0 \leq r < m$. This is where the Euclidean function (sometimes called a norm) comes in. Then, we may generalize the above division algorithm to hold for any nonzero $m = b, n = a \in R$, where $R$ is a general ring. The condition that our remainder is less than our divisor, $b$, is now replaced with the statement that $f(r) < f(b)$ or $r = 0$.