positivity in ordered ring

Theorem.

If  $(R,\le )$  is an ordered ring, then it contains a subset $R_{+}$ having the following :

• $R_{+}$ is under ring addition and, supposing that the ring contains no zero divisors, also under ring multiplication.

• Every element $r$ of $R$ satisfies exactly one of the conditions  $(1)r=0$,    $(2)r\in R_{+}$,    $(3)-r\in R_{+}$.

Proof.  We take  .  Let  $a,b\in R_{+}$.  Then   ,  , and therefore we have  ,  i.e.  $a+b\in R_{+}$.  If $R$ has no zero-divisors, then also  $ab\ne 0$  and  ,  i.e.  $ab\in R_{+}$.  Let $r$ be an arbitrary non-zero element of $R$.  Then we must have either    or    (not both) because $R$ is totally ordered.  The latter alternative gives that  .  The both cases that either  $r\in R_{+}$  or  $-r\in R_{+}$.