basic facts about ordered rings

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

Lemma 1.

If $a\mathrm{,}b\mathrm{,}c\mathrm{\in }R$ with , then .

Proof.

The contrapositive will be proven.

Let $a,b,c\in R$ with $a+c\ge b+c$. Note that $-c\in R$. Thus,

Lemma 2.

If $\mathrm{|}R\mathrm{|}\mathrm{\ne }\mathrm{1}$ and $R$ has a characteristic, then it must be $\mathrm{0}$.

Proof.

Suppose not. Let $n$ be a positive integer such that $\mathrm{char}R=n$. Since $|R|\ne 1$, it must be the case that $n>1$.

Let $r\in R$ with $r>0$. By the previous lemma, , a contradiction.∎

Lemma 3.

If $a\mathrm{,}b\mathrm{\in }R$ with $a\mathrm{\le }b$ and $c\mathrm{\in }R$ with , then $a\mathit{}c\mathrm{\ge }b\mathit{}c$.

Proof.

Note that $-c\in R$ and . Since $a\le b$, $-(ac)=a(-c)\le b(-c)=-(bc)$. Thus,

Lemma 4.

Suppose further that $R$ is a ring with multiplicative identity $\mathrm{1}\mathrm{\ne }\mathrm{0}$. Then .

Proof.

Suppose that . Since $R$ is an ordered ring, it must be the case that . By the previous lemma, $1\cdot 1\ge 0\cdot 1$. Thus, $1\ge 0$, a contradiction.∎