nLab totally ordered ring



(0,1)-Category theory



A totally ordered ring is an ordered ring whose order forms a total order.


This definition is adapted from Peter Freyd‘s definition of a totally ordered abelian group:

A totally ordered ring is an lattice-ordered ring RR such that for all elements aa in RR, a0a \leq 0 or a0-a \leq 0.

In a totally ordered ring, the join is usually called the maximum, while the meet is usually called the minimum

If the relation \leq is only a preorder, then the prelattice-ordered ring RR is said to be a totally preordered ring.


The integers, the rational numbers, and the real numbers are totally ordered rings.


  • Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

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