[MikeBattaglia]

It isn't true for all ordered rings, and the dual numbers are in fact a counterexample to the claim that it is true.

Beyond that I'm not sure what to tell you, other than it's fairly easy to see that the dual numbers do satisfy the axioms of an ordered ring that I gave. Here's a large survey of various infinitesimal systems by Philip Ehrlich where he also notes the dual numbers are an ordered ring: https://arxiv.org/pdf/1808.03345.pdf.