basic facts about ordered rings

Throughout this entry, (R,) is an ordered ring.

Lemma 1.

If a,b,cR with a<b, then a+c<b+c.


The contrapositive will be proven.

Let a,b,cR with a+cb+c. Note that -cR. Thus,


Lemma 2.

If |R|1 and R has a characteristicPlanetmathPlanetmath, then it must be 0.


Suppose not. Let n be a positive integer such that charR=n. Since |R|1, it must be the case that n>1.

Let rR with r>0. By the previous lemma, 0<rj=1n-1rj=1nr=0, a contradictionMathworldPlanetmathPlanetmath. ∎

Lemma 3.

If a,bR with ab and cR with c<0, then acbc.


Note that -cR and 0=c+(-c)<0+(-c)=-c. Since ab, -(ac)=a(-c)b(-c)=-(bc). Thus,


Lemma 4.

Suppose further that R is a ring with multiplicative identityPlanetmathPlanetmath 10. Then 0<1.


Suppose that 01. Since R is an ordered ring, it must be the case that 1<0. By the previous lemma, 1101. Thus, 10, a contradiction. ∎

Title basic facts about ordered rings
Canonical name BasicFactsAboutOrderedRings
Date of creation 2013-03-22 16:17:21
Last modified on 2013-03-22 16:17:21
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Result
Classification msc 06F25
Classification msc 12J15
Classification msc 13J25
Related topic MathbbCIsNotAnOrderedField