Idempotent Elements for Integer Multiplication

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.