Proving Properties of an Ordered Ring: R+ & R

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SUMMARY

The discussion centers on proving properties of an ordered ring, specifically that if R+ is well-ordered, then the minimum of R+ is 1 and that R is an integer ring. The participants emphasize the importance of establishing the existence of min(R+) and suggest assuming it is not 1 to derive a contradiction. This logical approach is essential for solving the problem effectively.

PREREQUISITES
  • Understanding of ordered rings and their properties
  • Familiarity with well-ordering principles
  • Knowledge of integer rings and their characteristics
  • Experience with proof techniques, particularly contradiction
NEXT STEPS
  • Study the properties of well-ordered sets in set theory
  • Learn about the structure and properties of integer rings
  • Explore proof techniques involving contradiction in mathematical logic
  • Investigate the implications of minimum elements in ordered sets
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Mathematicians, students studying abstract algebra, and anyone interested in the properties of ordered rings and their applications in mathematical proofs.

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Let R be an ordered Ring. Assume R+ is well-ordered
Prove:
a) min(R+) = 1.
b) R is an integer ring
 
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Sounds like a homework problem all right.

You didn't say it, but I assume you're looking for help? What have you done (successful or not), and where are you stuck?
 
(1) Why does min(R+) exist?
(2) Let u = min(R+), assume it is not 1, try to get a contradiction.
 

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