Properties_of_Ordered_Ring

Let $\struct {R, +, \circ, \le}$ 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 $x, y, z \in \struct {R, +, \circ, \le}$. Then the following properties hold $(1): \quad x < y \iff x + z < y + z$. Hence $x \le y \iff x + z \le y + z$ (because $\struct {R, +, \le}$ is an ordered group). $(2): \quad x < y \iff 0 < y + \paren {-x}$. Hence $x \le y \iff 0 \le y + \paren {-x}$ $(3): \quad 0 < x \iff \paren {-x} < 0$. Hence $0 \le x \iff \paren {-x} \le 0$ $(4): \quad x < 0 \iff 0 < \paren {-x}$. Hence $x \le 0 \iff 0 \le \paren {-x}$ $(5): \quad \forall n \in \Z_{>0}: x > 0 \implies n \cdot x > 0$ $(6): \quad x \le y, 0 \le z: x \circ z \le y \circ z, z \circ x \le z \circ y$ $(7): \quad x \le y, z \le 0: y \circ z \le x \circ z, z \circ y \le z \circ x$