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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?

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My first inclination would be to work out the case $K = k[T]/(T^3)$. –  Michael Joyce Mar 29 '12 at 4:24
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Could we perhaps call it something other than $K$? I keep thinking that it's a field and hence that I have an answer for this :) –  Dylan Moreland Mar 29 '12 at 5:14
    
Dear Soon, Your question is unclear. For example, what exactly is $V$? Is it a set of elements in $K^2$, or a set of maximal ideals in $K[x,y]$, or a set of prime ideals in $K[x,y]$, or ... ? Regards, –  Matt E Apr 30 '12 at 14:59

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