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.
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.