Order of Squares in Totally Ordered Ring without Proper Zero Divisors

Theorem
Let $\left({R, +, \circ, \le}\right)$ be a totally ordered ring whose zero is $0_R$ and whose unity is $1_R$.

Let $x, y \in \left({R, +, \circ, \le}\right)$ such that $0_R \le x, y$.

Then $x \le y \iff x \circ x \le y \circ y$.

When $R$ is one of the standard sets of numbers, i.e. $\Z, \Q, \R$, then this translates into:


 * If $x, y$ are positive then $x \le y \iff x^2 \le y^2$.

Note it does not hold for the complex numbers $\C$, as $\C$ cannot be given a total order compatible with its ring structure.

Proof

 * Assume $x \le y$.

As $\le$ is compatible with the ring structure of $\left({R, +, \circ, \le}\right)$, we have:


 * $x \ge 0 \implies x \circ x \le x \circ y$
 * $y \ge 0 \implies x \circ y \le y \circ y$

and thus as $\le$ is transitive, it follows that $x \circ x \le y \circ y$.


 * Now assume that $x \circ x \le y \circ y$.

Thus: