Find Idempotent Element in Z/mnZ (m,n Relatively Prime)

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SUMMARY

The discussion centers on demonstrating that the ring Z/mnZ, where m and n are relatively prime, contains an idempotent element other than 0 and 1. The key insight is derived from the equation a^2 - a = kmn for some k in Z, leading to the conclusion that bn is an idempotent element, where a and b satisfy the equation am + bn = 1. This realization confirms the existence of non-trivial idempotent elements in the specified ring.

PREREQUISITES
  • Understanding of ring theory and idempotent elements
  • Familiarity with coprime integers and their properties
  • Knowledge of the equation a^2 - a = kmn
  • Basic concepts of modular arithmetic in Z/mnZ
NEXT STEPS
  • Study the properties of idempotent elements in ring theory
  • Explore the implications of coprime integers in algebraic structures
  • Learn about the structure of Z/mnZ and its applications
  • Investigate the role of the equation am + bn = 1 in number theory
USEFUL FOR

Mathematicians, algebra students, and anyone interested in advanced ring theory and the properties of idempotent elements in modular arithmetic.

mansi
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here's a real tough one ( at least for me) ...show that the ring Z/mnZ where m ,n are relatively prime has an idempotent element other than 0 and 1.
i looked at examples and it works...
do we look for solutions of the equation a^2 -a = kmn , for some k in Z( that is, other than 0 and 1)?
help!
 
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m and n are coprime. The only thing you know about coprime integers is that there are numbers a and b such that am+bn=1. What can you conclude now?
 
ok...so am + bn= 1 implies...1-bn = am
that implies... bn(1-bn) = abmn = 0...
so bn is an idempotent element ...
i looked at " am +bn =1" a hundred times before posting this question...but it flashed just now! thanks a ton!
 

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