Can every polynomial generate an ideal?

Question

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal?

Helper parts

1. Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer Hilbert basis theorem? Consider $f\in R$, is $\langle f\rangle$ an ideal?

(Hilbert Basis Theorem). Every polynomial ideal in C[x] is finitely generated.

2. In contrast when $R$ is an infinite polynomial ring: does Hilbert Nullstellensatz explain things? Can every infinite polynomial generate an ideal?

Answer 1

Yes. More generally, let $R$ be a ring. For any $a\in R$, the subset $\langle a\rangle$ is the principal ideal of $R$ generated by $a$. This ideal is the smallest ideal of $R$ containing $a$.

A proof can be found here.

In your case, $R$ is some polynomial ring $R=A[x_1,\dotsc,x_n]$ for a ring $A$.