Archimedean

A ring R is said to be Archimedean if the ring is ordered, has a metric <math>| |</math> and for all <math>x,y</math> in R, x non-zero, there exists <math>n</math> in the natural numbers such that <math>n|x| > y </math>. Here concatentation with <math> n </math> denotes adding <math>n</math> times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the real numbers and the rational numbers. Examples of non-Archimedean are less simple.