This question is motivated by the following question: What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?
I'm wondering under what circumstances we can compute an ideal from its radical (assuming the ideal is not already equal to its radical). In trying to compute the primary decomposition of $(xy, x-yz)$ (see the linked question above), one has $\sqrt{I}=(x,y) \cap (x,z)$. Now, we cannot conclude $I=(x,y) \cap (x,z)$, since $(x,y) \cap (x,z)$ is not its own radical. So how can one compute $I$ from this?
The linked answer claims that we can write $I=Q_1 \cap Q_2$ with $\sqrt{Q_1}=(x,y)$ and $\sqrt{Q_2}=(x,z)$. I could not justify this and surely it is not true in general that $\sqrt{I}=P_1 \cap P_2$ implies $I=\tilde{P_1} \cap \tilde{P_2}$ with $\sqrt{\tilde{P_i}}=P_i$. Under what circumstances can we make this assumption?
Well, since $\sqrt{\tilde{P_i}^n}=P_i$, I fear that the only case when you can do the above assumption is when both ideals are trivial.