uniquess of hyperbolic numbers

Question

I'm trying to prove, the uniqueness of hyperbolic numbers, like the complex numbers are unique, but since hyperbolic numbers aren't a field, I can't use the ideas of this. Are there theorems of uniqueness of factor groups? Since (Hyperbolic numbers) $\mathbb{D} \cong \mathbb{R}[x]/\langle x^2-1\rangle$