Difference between revisions of "Archimedean ring"
From Encyclopedia of Mathematics
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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 | + | {{TEX|done}} |
| + | 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=18350
Archimedean ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_ring&oldid=18350
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article