An ordered ring is a (not necessarily commutative) ring with a total order $≤$ respecting the operations:
if $a ≤ b$ then $a + c ≤ b + c$,
if $0 ≤ a$ and $0 ≤ b$ then $0 ≤ ab$ (equivalently: if $a ≤ b$ and $0 ≤ c$, then $ac ≤ bc$).
An ordered ring is discretely ordered if there is no element between $0$ and $1$:
- if $0 ≤ a ≤ 1$ then $a = 0$ or $a = 1$.
gives an example of an ordered ring that is non-commutative. I do not have the competence to quickly tell whether that example is a discretely ordered ring (I would guess it is not).
Is there a non-commutative discretely ordered ring?