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I want to prove that "the spectrum of maximal ideals of a ring $A$ is a variety of $\mathbb{A}^n_k$ for some $n$ if and only if $A$ is a finitely generated $k$-algebra". I assume that $k$ is algebraically closed.

Any hints on how to make a start for each direction?

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    I assume $A$ is a commutative ring with a unity. $\mathbb{Z}_p$ is such a ring.2012-10-04

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The formulation of the question given by Andrew in the comments is meaningful but false (take $A = k[x]/x^2$). The correct statement comes from replacing "finitely generated $k$-algebra" with "finitely generated integral domain over $k$" and follows from the Nullstellensatz.

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    @Manos: no. Without further hypotheses, $A = k[x]/x^2$ is still a counterexample. The correct statement of the question is the following modification to Andrew's statement in the comments: _if_ $A$ is a finitely generated integral domain over $k$, then there is an affine variety with ring of regular functions $A$, and moreover the points of this affine variety can be identified with the maximal spectrum of $A$.$A$choice of embedding into affine space is one way to construct the desired variety (and depending on your definition of affine variety it may be the only way).2012-10-05