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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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If an algebraic set in affine n-space has a prime ideal then it is irreducible. (Hartshorne'...

I am confused about a step in Hartshorne's proof (final part of Corollary 1.4) that an algebraic set $Y$ in affine n-space $\mathbb{A}^n$ having a prime ideal $I(Y)$ in the polynomial ring over $n$ va …