Definition of ordered ring/field

Question

One way to define an ordered ring is as a ring with a total order $\leq$, satisfying

1. $x\leq y\implies x+z\leq y+z$;
2. $0\leq x$ and $0\leq y\implies 0\leq xy$.

Does it hurt to replace 1. with the condition

3. $x<y\implies x+z<y+z$?

What about for fields?

Answer 1

If you define $$x < y ⇔ (x ≤ y ~\text{and}~ x≠ y),$$ note that $$(A)~~ x = y ⇒ x + z = y + z \quad \text{and}\quad (B)~~x ≠ y ⇒ x + z ≠ y + z.$$ Now “(A) and (3) ⇒ (1)” and “(1) and (B) ⇒ (3)” are straightforward by case differentiation.

(A) is always true and (B) (being the converse to (A)) is true if addition is right-cancellable (which always holds for rings).

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If you think about semi-rings, consider $ℕ_0 = (ℕ_0,·,\max,≤)$ with the usual notion of “$≤$”. This is a semi-ring. Note that $0 < 1$, but $0·0 = 0·1$, where (1) and (2) certainly hold.

(If you want an multiplcative identity, extend by $∞$, defining $0·∞ = 0$ and $n·∞ = ∞$ else. This should work – not sure, though.)

Answer 2

Certainly, if you know (3), then (1) is an easy consequence. On the other hand, if you know (1), you might worry that you have a case when $x < y$ but $x + z = y + z$ (this is the only way you could possibly not satisfy (3)). But this is impossible---if $x + z = y + z$, then $x = y$, and therefore you couldn't have had $x < y$ (by virtue of $<$ being a total order).