(i) Define the radical of an ideal.

(ii) Assume the following statement: If $k$ is an algebraically closed field and $I \subseteq$ $k\left[x_{1}, \ldots, x_{n}\right]$ is an ideal, then either $I=(1)$ or $Z(I) \neq \emptyset$. Prove the Hilbert Nullstellensatz, namely that if $I \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ with $k$ algebraically closed, then

$$I(Z(I))=\sqrt{I}$$

(iii) Show that if $A$ is a commutative ring and $I, J \subseteq A$ are ideals, then

$$\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J} .$$

(iv) Is

$$\sqrt{I+J}=\sqrt{I}+\sqrt{J} ?$$

Give a proof or a counterexample.