Suppose $m, n > 1$ are positive integers which are relatively prime. Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. Two of them are $[0], [1]$, how will I find the other two?
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There are really only two other 'natural' guesses, aren't there? – pjs36 Apr 16 '15 at 17:38
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Hint: the equation $x^2 \equiv x$ is true mod $mn$ if and only if it is true mod $m$ and mod $n$.
Robert Israel
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$x$ is an idemptent element implies $mn$ divides $x(x-1)$ so if $mn$ divides $x$ you have $x=0$ and if $mn$ divides $x-1$ you have $x=1$ what if $n$ divides $x$ and $m$ divides $x-1$ you will find another and if you permte $m$ and $n$ you will find the fourth.
The list of the idemptent elements: $0,1,nu,mv,\cdots$ where $nu+mv=1$
Elaqqad
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