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I need to prove that a radical ideal in the coordinate ring $\mathbb{C}[V]$ (where V is an algebraic variety) is the intersection of all maximal ideals containing it.

The hint I was given was to consider the correspondence between ideals in a quotient ring $R/I$ and the ideals of $R$ containing $I$.

Any suggestions for where to begin? I don't even know where to start.

mkmath
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  • I will actually post a solution when I remember how this proof goes. (It has been a while, and I haven't slept well in about a week.) For now, though, I'll note that this statement above is probably true in general for any algebraically closed field rather than just $\mathbb{C}$. – Tanner Strunk Feb 21 '18 at 03:33
  • Still working out the rest of my answer, but I wanted to point out the geometry in the situation. You should also tag this thing with the algebraic-geometry tag, because that's really what this is. (The other tags are fine, too, of course, but you'll get more answers with including the algebraic-geometry tag.) – Tanner Strunk Feb 21 '18 at 03:38

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OK, so... I want to give a solution, but I also want to point out the geometry of this situation... the geometry is pretty important here. A radical ideal in $\mathbb{C}[V]$ corresponds to a (I think irreducible) subvariety of $V$. When we say maximal ideals containing $I$, what we really mean are maximal ideals corresponding to points in $\mathbb{C}^n$ (assuming our ambient dimension is $n$) which lie on this subvariety $V$. So the content of the statement, geometrically, is that a variety is made up of all the points contained in it as a set. (One ideal containing another corresponds to a reverse inclusion for their varieties. One maximal containing the radical $I$ corresponds to $V(I)$ containing the point corresponding to the maximal ideal.) So now we at least have some intuition for why the statement should be true if there's any justice in the world.

I was going to answer your algebraic content here, but apparently this question has already been answered with what I was meaning with Hilbert's Nullstellensatz basically: Intersection of all maximal ideals containing a given ideal I will be marking your question as a duplicate.

Tanner Strunk
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