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Lemma 1   If $a,b,c \in R$ with $a<b$ , then $a+c<b+c$ .

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 $|R| \neq 1$ and $R$ has a characteristic, then it must be $0$ .

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

Let $r \in R$ with $r>0$ . By the previous lemma, $\displaystyle 0<r \le \ldots \le \sum_{j=1}^{n-1} r \le \sum_{j=1}^n r=0$ , a contradiction.

Lemma 3   If $a,b \in R$ with $a \le b$ and $c \in R$ with $c<0$ , then $ac \ge bc$ .

Proof. Note that $-c \in R$ and $0=c+(-c)<0+(-c)=-c$ . Since $a \le b$ , $-(ac)=a(-c) \le b(-c)=-(bc)$ . Thus,

Lemma 4   Suppose further that $R$ is a ring with multiplicative identity $1 \neq 0$ . Then $0<1$ .

Proof. Suppose that $0 \not< 1$ . Since $R$ is an ordered ring, it must be the case that $1<0$ . By the previous lemma, $1 \cdot 1 \ge 0 \cdot 1$ . Thus, $1 \ge 0$ , a contradiction.

basic facts about ordered rings is owned by Warren Buck.