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basic facts about ordered rings
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(Result)
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Throughout this entry, is an ordered ring.
Proof. The contrapositive will be proven.
Let
with
. Note that . Thus,

Proof. Suppose not. Let  be a positive integer such that
 . Since
 , it must be the case that  .
Let with . By the previous lemma,
, a contradiction. 
Lemma 3 If with and with , then .
Proof. Note that  and
 . Since  ,
 . Thus,

Proof. Suppose that  . Since  is an ordered ring, it must be the case that  . By the previous lemma,
 . Thus,  , a contradiction. 
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"basic facts about ordered rings" is owned by Wkbj79.
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(view preamble)
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 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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Pending Errata and Addenda
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