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I have a coordinate ring and i have to find ideals: one radical not prime, prime not maximal and maximal. I have some ideas but i'm unsure. The co-ordinate ring is $\mathbb{C}[x,y,z,]/ I(A)$ where $A=V(xz,xy,yz)$

For the radical one, i thought the ideal $(xy,xz,yz)$ which is radical by the Nullstellensatz?

For the prime but not maximal one, i thought an ideal generated by a single of the variables, i.e. $(x)$

For the maximal one, i know any maximal one of $\mathbb{C}[x,y,z]$ will have form $(x-a,y-b,z-c)$ so i thought this would work for the coordinate ring also.

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The ideals in the coordinate ring are exactly the ideals in the polynomial ring which contain the ideal $I$.

In order for your construction of a maximal ideal to work, you therefore have to choose the maximal ideal so that it contains $I$; this is equivalent to requiring that the point that it corresponds to lies on the variety corresponding to $I$.

The construction that you suggest for a prime ideal does not work as such, because the ideal in the coordinate ring generated by $x$ is not prime. This is because, in the coordinate ring, $yz=0$. So $yz$ is in the ideal generated by $x$, but neither $y$ nor $z$ is.

For the radical ideal, sure, what you propose works.

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