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[parent] basic facts about ordered rings (Result)

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

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,

\begin{displaymath}\begin{array}{rl} b & =b+0 \ & =b+c+(-c) \ & \le a+c+(-c) \ & =a+0 \ & =a. \qedhere \end{array}\end{displaymath}
$ \qedsymbol$
Lemma 2   If $ \vert R\vert \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 $ \vert R\vert \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. $ \qedsymbol$

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,
\begin{displaymath}\begin{array}{rl} bc & =bc+0 \ & =bc+(ac+(-(ac))) \ & =(b... ... & =(-(bc)+bc)+ac \ & =0+ac \ & =ac. \qedhere \end{array}\end{displaymath}
$ \qedsymbol$
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. $ \qedsymbol$



"basic facts about ordered rings" is owned by Wkbj79.
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See Also: $\mathbb{C}$ is not an ordered field


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Cross-references: multiplicative identity, ring, contradiction, integer, positive, characteristic, contrapositive, ordered ring
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This is version 8 of basic facts about ordered rings, born on 2006-10-01, modified 2007-09-22.
Object id is 8405, canonical name is BasicFactsAboutOrderedRings.
Accessed 889 times total.

Classification:
AMS MSC: 13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings)
 12J15 (Field theory and polynomials :: Topological fields :: Ordered fields)
 06F25 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered rings, algebras, modules)
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