This has been bugging me for a while.


According to <https://en.wikipedia.org/wiki/Euclidean_division>, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r < b$. 


Furthermore, for any Euclidean domain $R$, division with remainder can be defined as follows:
$a = t b + r$, and either $r = 0$ or $f(r) < f(b)$.
where $f$ is the euclidean function of $R$. $\mathbb{Z}$ fits into this classification by letting $f(n) = |n|$. 


My problem with this "generalization" is that $t$ and $r$ need not be unique. Is there a classification for structures which support division with remainder where $t$ and $r$ are unique?



What structure supports division to a unique quotient and remainder?
In [M. A. Jodeit, Jr., Uniqueness in the division algorithm, The American Mathematical Monthly 74 (1967), 835–836] it is shown that a Euclidean Domain in which quotient and remainder are strictly unique (for the integers there is a choice of sign) is either a field or a polynomial ring over a field.


