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 - but for as follows: $a general Euclidean domain= t b + r$, 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 $r$ as defined in the above example are need not be unique. Is there a classification for structures which support division with remainder where $t$ and $r$ are unique?
