Idempotents which are products of two nilpotents

Abstract Over any GCD (greatest common divisor) commutative domain we show that the nontrivial 2 × 2 2\times 2 idempotent matrices are products of two nilpotent matrices. In order to find explicitly such decompositions, two procedures are described. Assisted by computer, we were able to find an example of idempotent 2 × 2 2\times 2 matrix over Z [ − 5 ] {\mathbb{Z}}\left[\sqrt{-5}] , which shows that the GCD condition is (sufficient but) not necessary. Finally, a generalization is discussed and some open questions stated.