Consider the equations $ x^3=0 , y^3=0 , xy(x+y)=0 $ where the polynomials live in $K[x,y]$ where K is a k-algebra (k field). Let V be the points that vanish on all this polynomials. Consider the ideal $I(V)$ of all the polynomials in $K[x,y]$ that vanish in V. Find the ideal $I(V)$ . Does $x+y$ belong to $I(V)$?
I this problem I have no idea what can I do. We allow K to have zero divisors. Otherwise the problem it´s very easy. What can I do?