PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
[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,

$\begin{array}{rl} b & =b+0 \\ & =b+c+(-c) \\ & \le a+c+(-c) \\ & =a+0 \\ & =a. \qedhere \end{array}$
$ \qedsymbol$
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. $ \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,
$\begin{array}{rl} bc & =bc+0 \\ & =bc+(ac+(-(ac))) \\ & =(bc+ac)+(-(ac)) \\ & \le (bc+ac)+(-(bc)) \\ & =-(bc)+(bc+ac) \\ & =(-(bc)+bc)+ac \\ & =0+ac \\ & =ac. \qedhere \end{array}$
$ \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.
(view preamble | get metadata)

View style:

See Also: $\mathbb{C}$ is not an ordered field


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: multiplicative identity, ring, contradiction, integer, positive, characteristic, contrapositive, ordered ring
There is 1 reference to this entry.

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 1353 times total.

Classification:
AMS MSC13J25 (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)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)