Idempotent Elements for Integer Multiplication

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Theorem

There are exactly two integers which are idempotent with respect to multiplication:

$0 \times 0 = 0$
$1 \times 1 = 1$


Proof

The integers $\left({\Z, +, \times}\right)$ form an integral domain.

By definition of integral domain, therefore, the integers form a ring with no (proper) zero divisors.


The result follows from Idempotent Elements of Ring with No Proper Zero Divisors.

$\blacksquare$


Sources