Properties of Ordered Ring

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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, $0_R < 1_R$

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

Proof

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Sources

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

This article, or a section of it, is suspected of being a duplicate of an existing article. In particular: This duplicates, at least in part with Properties of Totally Ordered Field You can help ProofWiki by finding that page, and either merging this one with it or marking either for deletion. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Duplicate}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.