Properties of Ordered Ring

Theorem
Let $$\left({R, +, \circ, \le}\right)$$ be an ordered ring whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$U_R$$ be the group of units of $$R$$.

Let $$x, y, z \in \left({R, +, \circ, \le}\right)$$.

Then the following properties hold:


 * $$(1) \quad x < y \iff x + z < y + z$$. Hence $$x \le y \iff x + z \le y + z$$


 * $$(2) \quad x < y \iff 0 < y + \left({-x}\right)$$. Hence $$x \le y \iff 0 \le y + \left({-x}\right)$$


 * $$(3) \quad 0 < x \iff \left({-x}\right) < 0$$. Hence $$0 \le x \iff \left({-x}\right) \le 0$$


 * $$(4) \quad x < 0 \iff 0 < \left({-x}\right)$$. Hence $$x \le 0 \iff 0 \le \left({-x}\right)$$


 * $$(5) \quad \forall n \in \N^*: x > 0 \implies n \cdot x > 0$$


 * $$(6) \quad x \le y, 0 \le z: x \circ z \le y \circ z, z \circ x \le z \circ y$$


 * $$(7) \quad x \le y, z \le 0: y \circ z \le x \circ z, z \circ y \le z \circ x$$

Total Ordering
If, in addition, $$\left({R, +, \circ, \le}\right)$$ is totally ordered, the following properties also hold:


 * $$(8) \quad 0 < x \circ y \implies \left({0 < x \and 0 < y}\right) \or \left({x < 0 \and y < 0}\right)$$


 * $$(9) \quad x \circ y < 0 \implies \left({0 < x \and y < 0}\right) \or \left({x < 0 \and 0 < y}\right)$$


 * $$(10) \quad 0 \le x \circ x$$. In particular, if $$R$$ has a unity, $$0_R < 1_R$$


 * $$(11) \quad x \in U_R \implies 0 < x \iff 0 < x^{-1}, x \le 0 \iff x^{-1} \le 0$$