Idempotent elements

  • Thread starter mansi
  • Start date
  • #1
61
0
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!
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
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?
 
  • #3
61
0
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!
 

Related Threads on Idempotent elements

  • Last Post
Replies
9
Views
16K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
2
Views
8K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
Top