I am reading materials about algebraic geometry. I know some commutative algebra but definitely not enough. In AG, many objects are defined in prime ideals.
For example, for any commuative ring $A$, $SpecA$ is the set of prime ideals of $A$.
Also, let $S$ be a subset of $A$, $$V(S) := {[p]∈SpecA: S⊆p}$$
I wonder the existence of the prime ideals. Is there a theorem in commutative algebra that ensures this?
Question: "I wonder the existence of the prime ideals. Is there a theorem in commutative algebra that ensures this?"
Answer: If $k$ is a field it follows by the Hilbert Basis Theorem that $A:=k[x_1,..,x_n]/I$ is Noetherian. Hence any chain of ideals $I_1 \subseteq I_2 \subseteq \cdots $ must terminate. This proves the existence of maximal ideals in $A$ which are prime. Hence if your ring is a finitely generated commutative $k$-algebra you do not need Zorns lemma/Axiom of choice to prove existence.
