## Ordered Fields

A **ring** is a set $R$ with the usual operations of multiplication and addition from $R \times R \rightarrow R$ which satisfy the properties of:

- associativity under addition
- commutativity under addition
- the existence of additive identity
- the existence of an additive inverse
- associativity under multiplication
- distributivity across addition

These properties are defined in the above mentioned link to the page real number axioms.

If the additional property of commutativity under multiplication holds then that ring is called a **commutative ring**. If there exists a multiplicative identity then it is called a **ring with unit**. A **field** is a commutive ring with unit with the existence of a multiplicative inverse for each element in $R$ except the element 0. A field allows for the existence of fractions since any fraction $\frac{a}{b}$ can be written in terms of an multiplicative inverse such as $\frac{a}{b}^{-1}.$ The usual method of fractional addition and multiplication hold in a field.

A field is said to be an **ordered field** if there exists a subset $P$ of $R$ such that

- $0 \notin P$
- if $a \neq 0$, then $a$ or $-a$ is an element of $P$ (but not both)
- if a and b are elements of $P$, then $a+b$ and $ab$ are elements of $P$

It should now be apparent that $P$ is the set of positive numbers. An ordered ring allows for the existence of the $<$ relation and many of the properties.