Algebraic Geometry

TL;DR: The Z(S) set is equal to the Z(〈S〉) set for any subset S of a ring A.

Abstract: Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉 consists of combinations of elements in S, we have Z(S) ⊆ Z(〈S〉). Since S ⊆ 〈S〉, we have Z(〈S〉) ⊆ Z(S) as well.