Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ?

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

1 Answer 1

up vote 1 down vote accepted

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.