EDB — 1ZT

[1ZT] Suppose that in a ring \(A\) there is a total ordering \(≤\) such that for every \(x, y, z ∈ A\) you have \(x ≤ y ⇒ x + z ≤ y + z\); then show that these are equivalent

• \(x ≤ y \, ∧\, 0 ≤ z \quad ⇒\quad x · z ≤ y · z\);

• \(x≥ 0∧ y ≥ 0 \quad ⇒\quad x · y ≥ 0 \) .