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) $$x < y \iff x + z < y + z$$. Hence $$x \le y \iff x + z \le y + z$$.
 * 2) $$x < y \iff 0 < y + \left({-x}\right)$$. Hence $$x \le y \iff 0 \le y + \left({-x}\right)$$.
 * 3) $$0 < x \iff \left({-x}\right) < 0$$. Hence $$0 \le x \iff \left({-x}\right) \le 0$$.
 * 4) $$x < 0 \iff 0 < \left({-x}\right)$$. Hence $$x \le 0 \iff 0 \le \left({-x}\right)$$.
 * 5) $$x \le y, 0 \le z: x \circ z \le y \circ z, z \circ x \le z \circ y$$.
 * 6) $$x \le y, z \le 0: y \circ z \le x \circ z, z \circ y \le z \circ x$$.

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


 * 1) $$0 < x \circ y \implies \left({0 < x \land 0 < y}\right) \lor \left({x < 0 \land y < 0}\right)$$.
 * 2) $$x \circ y < 0 \implies \left({0 < x \land y < 0}\right) \lor \left({x < 0 \land 0 < y}\right)$$.
 * 3) $$0 \le x \circ x$$. In particular, if $$R$$ has a unity, $$0_R < 1_R$$.
 * 4) If $$x \in U_R$$, then $$0 < x \iff 0 < x^{-1}, x \le 0 \iff x^{-1} \le 0$$.