### Table of Contents

## Ordered rings

Abbreviation: **ORng**

### Definition

An \emph{ordered ring} is a structure $\mathbf{A}=\langle A,+,-,0,\cdot,1,\le\rangle$ such that

$\langle A,+,-,0,\cdot,1\rangle$ is a ring

$\langle A,\le\rangle$ is a linear order

$+$ is \emph{order-preserving}: $x\le y\Longrightarrow x+z\le y+z$

$\cdot$ is \emph{order-preserving} for positive elements: $x\le y\text{ and }0\le z\Longrightarrow xz\le yz$

Remark: This is a template. If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \ldots y)=h(x) \ldots h(y)$

### Definition

An \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

None

### Subclasses

[[Complete ordered rings]]

[[Ordered fields]]

### Superclasses

[[Abelian ordered groups]] reduced type

[[Ordered monoids]] reduced type

[[Rings]] reduced type