Invertible elements in the number line $R$

Question

In SDG the geometric line is described by the number line $R$ which is a not a field, but a ring: this implies some of its elements are not invertible. One of the key idea is the set $D=[x:x^2=0]$ where $x$ is not necessarily zero. My question is, are these $x\in D$ the only elements $\in R$ which are not invertible?

Answer 1

The invertible elements of the dual numbers $\mathbb R[x]/(x^2)$ are exactly things whose representatives have nonzero constant term.

Everything has a representative that looks like $\alpha +\beta x +(x^2)$, so the invertible elements are exactly the things with $\alpha \neq 0$.