Ordered rings are (usually commutative) rings which have an additional structure, a linear order compatible with the ring structure. This tag is for questions regarding ordered rings and their properties, as well proofs related to un-orderability of certain rings.
Let $R$ be a (usually commutative) ring, then $R$ is an ordered ring if there is a binary relation $\leq$ on $R$ with the following properties:
- $\leq$ is a linear (total) order $R$.
- If $x\leq y$ then $x+z\leq y+z$.
- If $0\leq x,y$ then $0\leq ab$.
For example: $\Bbb N$, $\Bbb Z$, $\Bbb Q$ and $\Bbb R$ are ordered rings.
If $R$ is an ordered ring then all the squares are positive, and so are their sums. Since $-1$ is not positive we have that in an ordered ring, $-1$ is not the sum of squares, and therefore rings like $\Bbb Z[i]$ or $\Bbb C$ cannot be consideres as ordered rings.