@Phosphenes @johnpaulflintoff @pluralistic
It’s a fun exercise because it forced me to think how to show this without resorting to flashy terms like “algebraic rings”.
In your algebra, some of the following axioms have to be false:
-x = 0 - x
x * y = y * x
x * (a - b) = x*a - x*b
Because if all are true:
-x * -y = (0 - x) * (0 - y) = (0 - x) * 0 - (0 - x) * y = 0 - (0 - x) * y = 0 - y * (0 - x) = 0 - (y * 0 - y * x) = 0 - 0 + y * x = y * x = x * y