Definition:Affine Algebraic Set

Definition
Let $k$ be a field.

Let $A = k[X_1,\ldots,X_n]$ be the Ring of Polynomial Functions in $n$ variables over $k$.

Then a subset $X \subseteq k^n$ is an affine algebraic set if there exists a set $J \subseteq A$ such that


 * $X = \{ x \in k^n : f(x) = 0,\ \forall f \in J \} $