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I'm looking for examples of rings with the property that for each $P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)

† other than domains!

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1  
One is $\mathbb{C}[x]/(x(x-1))$. – Youngsu May 29 '13 at 18:25
2  
Given $P$, it's quite unclear whether you're asking that $a$ be idempotent for every $a$ such that $P=Ann(a)$ or that $a$ be idempotent for some $a$ such that $P=Ann(a)$. If the latter, take the direct sum of two domains. – Steven Landsburg May 29 '13 at 18:25
    
@Steven Landsburg:The latter case is what I mean:$P=Ann_R(a)$ for some idempotent $a$. – QED May 30 '13 at 10:13

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