Example

In $\mathbf{Z}$, if we take $P$ to be the set of positive (in the usual sense) even integers, we get a ring which is not totally ordered.

Recall also that, in a totally ordered abelian group, the relation $n \cdot x = 0$ (for a natural number $n \neq 0$) implies $x = 0$ (VI, p. 4); this gives us the following result.