An ordered ring is a ring $R=(X,\oplus ,\otimes )$ with a total order $\le$ compatible with the ring structure. Specifically, it must satisfy these axioms for any $a,b,c\in X$:

• If $a\le b$, then $a\oplus c\le b\oplus c$.
• If $0\le a$ and $0\le b$, then $0\le a\otimes b$

An element $a$ of the ring is called "positive" if $0<a$ and "negative" if $a<0$. The second axiom, then, says that the product of nonnegative elements is nonnegative.

An ordered ring that is also a field is an ordered_field.

Basic Properties

• For any element $a$, $a\le 0$ if and only if $0\le -a$.

• The product of nonpositive elements is nonnegative.

• The square of any element is nonnegative.

• The additive identity $1\ge 0$. (Unless the ring is trivial, $1>0$.)

Examples

The real numbers are an ordered ring (in fact, an ordered_field), as is any subring of $R$, such as $Q$.

The complex numbers are not an ordered ring, because there is no way to define the order between $0$ and $i$. Suppose that $0\le i$, then, we have $0\le i\times i=-1$, which is false. Suppose that $i\le 0$, then $0=i+(-i)\le 0+(-i)$, but then we have $0\le (-i)\times (-i)=-1$, which is again false. Alternatively, $i^2=-1$ is a square, so it must be nonnegative; that is, $0\le -1$, which is a contradiction.