Q:

Intersection of infinitely many prime ideals

I was thinking of the fact that, if you take an infinite number of prime ideals $p_i\mathbb{Z}$ in the ring $\mathbb{Z}$, then their intersection is zero. What kind of commutative rings satisfy that the intersection of an infinite number of prime ideals is zero?
Is this true for any integral domain?

A:

Take $R=\mathbb{C}[x,y]$, polynomial ring in two variables. Consider maximal ideals $P_a=(x-a,y)$ where $a\in\mathbb{C}$. Then $\cap_a P_a=(y)\neq 0$.

A:

This is not true for any integral domain. Consider a polynomial ring in countably infinitely many variables: $k[x_1,x_2,\dots]$, and consider the ideals $I_1 = (x_1,x_2,\dots)$, $I_2 = (x_1, x_3, x_4,\dots)$, and so on. In other words, $I_k$ is generated by $x_1$ and $x_i$ for all $i>k$. Then the intersection of all $I_k$'s is just $(x_1)$. 

