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The Nullstellensatz for $\mathbb{C}[x_1, \ldots, x_n]$ gives a dictionary between radical ideals and varieties, which makes the following assertion obvious: a radical ideal in $\mathbb{C}[x_1, \ldots, x_n]$ is the intersection of the maximal ideals containing it. Is there a simple purely algebraic way of seeing this?

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Any radical ideal (in any ring) is an intersection of prime ideals. So the question becomes why a prime ideal is an intersection of maximal ideals. Such rings are called Jacobson rings. A field is clearly Jacobson. It is a general fact (a more general form of the Nullstellensatz) that a ring finitely generated over a Jacobson ring is Jacobson. (This is in Eisenbud ch. 4.) So any affine coordinate ring is Jacobson.

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    Thanks. I was wondering if there was a really simple way, but it seems not.2011-03-04
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There is an elegant abstract approach to the Hilbert Nullstellensatz discovered circa 1950 by Goldman, Krull and Zariski. For more on this look up the terms: G-domain, Hilbert ring, Jacobson ring, e.g. see Kaplansky's textbook Commutative Rings. See also D.J. Bernstein's ZGK page which has some online links, including expositions of an "elementary" proof by R. Munshi (1999).