Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$:


1. $I = \langle (c\_i) \rangle$ is generated by Casimirs $c\_i$, i.e. each $c\_i\in Z$
2. $I$ is a Poisson ideal, i.e. $\{I,R\} \leq I$.


I'm having trouble distinguishing these geometrically: both seem to be about subschemes of $Spec\ R$ that are unions of symplectic leaves. But that seems too good to be true (or I would hope someone would have told me already). Still, I will ask:



> 
> If $I$ is a Poisson ideal, is $\sqrt{I} = \sqrt{ \langle I \cap Z \rangle}?$
> 
> 
> 


If so, what is a reference? If not, what is a counterexample?



Poisson ideals vs. ideals generated by Poisson central elements
I am not that familiar with the algebraic approach but the intersection on the rhs seems quite hard to me. Take the Poisson structure on the plane given by $\{x,y\}=xy$. Then $Z$ contains only constant functions and therefore for any Poisson ideal $I$ you have $I\cap Z$ equal to $Z$ or $0$ depending whether the ideal contains constants or not. Take $I=\langle x,y\rangle$ and it looks to me your condition is not fulfilled...


To put it another way there seems to me to be many situations in which the Poisson center is given only by constants and still the symplectic foliation is quite rich, thus many Poisson ideals.


