# Idempotent/Spec

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 ?

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Note that $(1,0)\cdot(1,0)=(1,0)$ in $B\times C$ but $\neq (1,1),(0,0)$. –  Pedro Tamaroff Mar 31 '14 at 14:01
Ok but what about the implication $1 \to 2$ ? –  WLOG Mar 31 '14 at 18:01

## 1 Answer

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 $R=eR\oplus (1-e)R$.

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