I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise:
$A$ is a commutative ring; show this $3$ conditions are equivalent:
1) $A$ contains a non-trivial idempotent
2) $A \cong B \times C $ for some nonzero rings $B$ and $C$
3) Spec($A$) with the Zariski topology is not connected
Any hint ?