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 $\struct {\Z, +, \times}$ 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

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.5$