Let $F$ be a field and let $I$ and $J$ be ideals in a ring $F[x_1,...,x_n]$.
Generally, $\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I}+\sqrt{J}$.

Give an example in $F[x_1,...,x_n]$ where $\sqrt{I+J} \ne \sqrt{I}+\sqrt{J}$.

2. Originally Posted by KaKa

Let $F$ be a field and let $I$ and $J$ be ideals in a ring $F[x_1,...,x_n]$.
Generally, $\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I}+\sqrt{J}$. false!
who told you that? for example in $\mathbb{R}[x,y]$ let $I=, \ J=.$ then both $I,J$ are prime ideals and hence $\sqrt{I}=I, \sqrt{J}=J$ and so $\sqrt{I}+\sqrt{J}=I+J=.$

but $\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I+J}=\sqrt{} = \neq .$

Give an example in $F[x_1,...,x_n]$ where $\sqrt{I+J} \ne \sqrt{I}+\sqrt{J}$.
same as above.