I have stumbled across the following exercise on radicals of ideals of rings. I shall show that: $\operatorname{rad}(x+y^2,x^2+2xy^2)$ is a maximal ideal of $\mathbb{C}[x,y]$, but $(x+y^2,x^2+2xy^2)$ is not. How can I show this?
Also, is the radical of a prime ideal equal to the ideal itself? so does $\operatorname{rad}(p)=p$ hold?
And a last question, I shall find a ring $R$ with exactly 17 ideals. My first idea was to take something similar to $\mathbb{Z}/2^{17}\mathbb{Z}$, where the ideals are $(0), (\mathbb{Z}/2^i\mathbb{Z})$ for any $i\le17$. But are those then the only ideals? Is not the union of one ideal with the zero ideal a new ideal? Then it would be difficult to find a Ring with the required number of ideals, wouldn't it?
Thanks in advance for any help!