Idempotent/Spec

Question

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 ?

Answer 1

This hint should help you with all parts:

For any idempotent $e$ in a commutative ring, $eR$ and $(1-e)R$ are also rings with identities $e$ and $1-e$ respectively. Moreover they are ideals in $R$ and $R=eR\oplus (1-e)R$.