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Difference between revisions of "Archimedean ring"

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A partially [[Ordered ring|ordered ring]] the additive group of which is an [[Archimedean group|Archimedean group]] with respect to the given order. An Archimedean totally ordered ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013130/a0131301.png" /> is either a ring with zero multiplication (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013130/a0131302.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013130/a0131303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013130/a0131304.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013130/a0131305.png" />) 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.
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A partially [[Ordered ring|ordered ring]] the additive group of which is an [[Archimedean group|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>

Latest revision as of 17:56, 11 April 2014

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://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