## 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:

3. the existence of additive identity
4. the existence of an additive inverse
5. associativity under multiplication
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
1. $0 \notin P$
2. if $a \neq 0$, then $a$ or $-a$ is an element of $P$ (but not both)
3. 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.