Definition:Affine Coordinate Ring

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Definition

Let $k$ be a field.

Let $Y \subseteq k^n$ be an affine algebraic set.

Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.

Let $\map I Y \subseteq k \sqbrk {X_1, \ldots, X_n}$ be the vanishing ideal of $Y$.


Then the affine coordinate ring of $Y$ is defined as the quotient ring:

$\ds \map A Y := k \sqbrk {X_1, \ldots, X_n}/\map I Y$


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