# Properties of Ordered Ring

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of basic complexity.In particular: This could do with being split up into (count 'em) 11 transcluded subpages, each with its own little theorem on it. Lots of this stuff has already been covered, so it would just be a matter of finding them. This will be a task for a very long, wet, miserable weekend in which I have no other projects on the go.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 `{{Refactor}}` from the code. |

## 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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.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 `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 FieldYou 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. |