Is there a name for a module with Euclidean division?

I have a module (over the integers, if necessary) on which I can define a total order.

As an example, my order might be defined by: $(x_1, ldots, x_n) < (y_1, ldots, y_n)$ iff $min(x_i) < min(y_i)$.

This allows me to define Euclidean division by implementing the basic algorithm:

  1. Find the least $n$ such that $X – nY – Y < 0$ by iteration. (Handle $Y < 0$ by negation.)
  2. Call $n$ the quotient and $X – nY$ the remainder.

Is this type of module something well-studied with a name I can use to find relevant material? Do I need to tighten any of my axioms to make the above statements generally true? Is there perhaps a more elegant generalisation of my axioms which is well-studied?

Excuse my rustiness, it’s been a few years since I studied algebra formally.

submitted by /u/jaakhaamer
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Nevin Manimala

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