You are here
Homepositivity in ordered ring
Primary tabs
positivity in ordered ring
Theorem.
If $(R,\,\leq)$ is an ordered ring, then it contains a subset $R_{+}$ having the following properties:

$R_{+}$ is closed under ring addition and, supposing that the ring contains no zero divisors, also under ring multiplication.

Every element $r$ of $R$ satisfies exactly one of the conditions $(1)\,\,r=0$, $(2)\,\,r\in R_{+}$, $(3)\,\,r\in R_{+}$.
Proof. We take $R_{+}=\{r\in R:\,\,0<r\}=\{r\in R:\,\,0\leq r\,\wedge\,0\neq r\}$. Let $a,\,b\in R_{+}$. Then $0<a$, $0<b$, and therefore we have $0<a\!+\!0<a\!+\!b$, i.e. $a\!+\!b\in R_{+}$. If $R$ has no zerodivisors, then also $ab\neq 0$ and $0=a0<ab$, i.e. $ab\in R_{+}$. Let $r$ be an arbitrary nonzero element of $R$. Then we must have either $0<r$ or $r<0$ (not both) because $R$ is totally ordered. The latter alternative gives that $0=r\!+\!r<r\!+\!0=r$. The both cases mean that either $r\in R_{+}$ or $r\in R_{+}$.
Mathematics Subject Classification
06F25 no label found12J15 no label found13J25 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections