Properties of Ordered Ring

Theorem

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 $x \le 0 \iff 0 \le \paren {-x}$

$(5): \quad \forall n \in \Z_{>0}: x > 0 \implies n \cdot x > 0$

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

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

Total Ordering

If, in addition, $\struct {R, +, \circ, \le}$ is totally ordered, the following properties also hold:

$(8): \quad 0 < x \circ y \implies \paren {0 < x \land 0 < y} \lor \paren {x < 0 \land y < 0}$

$(9): \quad x \circ y < 0 \implies \paren {0 < x \land y < 0} \lor \paren {x < 0 \land 0 < y}$

$(10): \quad 0 \le x \circ x$. In particular, if $R$ is non-null and has a unity, $0_R < 1_R$

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.11$