Archimedean ring

A partially ordered ring the additive group of which is an Archimedean group with respect to the given order. An Archimedean totally ordered ring $R$ is either a ring with zero multiplication (i.e. $xy=0$ for all $x$ and $y$ in $R$) over an additive group which is isomorphic to some subgroup of the group of real numbers, or else is isomorphic to a unique subring of the field of real numbers, taken with the usual order. An Archimedean totally ordered ring is always associative and commutative.