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:
- Find the least $n$ such that $X – nY – Y < 0$ by iteration. (Handle $Y < 0$ by negation.)
- 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.