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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?

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# What structure supports division to a unique quotient and remainder?

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, division with remainder can be defined - but for a general Euclidean domain, $t$ and $r$ as defined in the above example are not unique.

Is there a classification for structures which support division with remainder where $t$ and $r$ are unique?