# Archimedean ring

From Encyclopedia of Mathematics

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.

#### References

[1] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |

**How to Cite This Entry:**

Archimedean ring.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Archimedean_ring&oldid=31540

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article