Is there a non-trivial ordered ring with an "integer-esque" modulo function?

Question

(I'm inspired by this question.)


Is there a [not-necessarily-commutative non-simple ordered ring with a 1 that's not equal to 0]
which is not isomorphic to the integers but is such that

for all positive elements m, for all elements x, there is a least
non-negative y such that x-y is in the (two-sided) ideal generated by m

?


(An isomorphism between an ordered ring and the integers is automatically order-preserving.)

Answer 1

I think the ring of rational polynomials with integer constant coefficient should do. (Ordered in the natural way by the sign of the leading coefficient).