My question is pretty easy, however I got stuck to get the answer.
Assume the polynomial ring $ R=\mathbb Z[x,y]$ and $I=(xy)$ be an ideal of $R$. I want to compute the radical of $I$.
I am trying to show that $I$ is prime ideal since I know tha radical for prime ideal will be the prime ideal itself. Anyway, what is the radical of $I$.
Any help will be appreciated.
If $P\in Rad(I)$ then there exists $k\in\mathbb{N}$ and $Q\in\mathbb{Z}[x,y]$ such that $P^k=xyQ$. And obviously, $k>0$.
Consequently, $x$ and $y$ divide $P^k$.
But $x$ and $y$ are irreducible. Hence, by Euclid's lemma, each of them must divide $P$.
Finally, since $x$ and $y$ are coprime, their lcm (namely $xy$) must divide $P$. So $P\in I$.
Hoping it's correct ...