Note on lattice-ordered rings with positive squares

Bourbaki, Birkhoff-Pierce and Fuchs pointed out or showed that a lattice-ordered field in which each square is positive must be totally ordered. Yang proved that a lattice-ordered ring R is a totally ordered skew-field if and only if every strictly positive element of R is invertible and each square in R is positive. In this note, we construct a simple example to explain the difference between order-isomorphisms and lattice-isomorphisms, and show that the difference can be dropped in lattice-ordered rings. Especially, this yields an extension and an elementary proof of the main theorem in Yang's paper.