Prove that every finitely generated ideal \[ I\subset\mathcal O(\mathbb C) \] is principal.
Solution
1. Every ideal generated by at most one entire function is principal.
The zero ideal is \((0)\), and an ideal generated by one function \(f\) is \((f)\) by definition.
2. Let \(f,g\in\mathcal O(\mathbb C)\) be nonzero. There exists an entirefunction \(d\) whose zero order at each \(a\in\mathbb C\) is\[ \operatorname{ord}_a(d) =\min\{\operatorname{ord}_a(f),\operatorname{ord}_a(g)\}. \]
The common zero set of \(f\) and \(g\) is discrete. At each common zero prescribe the positive integer multiplicity given by the displayed minimum. The Weierstrass product theorem produces an entire function having exactly these zeros with exactly these multiplicities. If there are no common zeros, take \(d=1\).
3. The quotients\[ F=\frac f d, \qquad G=\frac g d \]are entire and have no common zeros.
At every zero of \(d\), its multiplicity is at most the zero multiplicity of both \(f\) and \(g\), so both quotients have removable singularities there and extend to entire functions.
If \(F\) and \(G\) vanished at the same point \(a\), then both \(f\) and \(g\) would vanish there to order strictly larger than \(\operatorname{ord}_a(d)\), contradicting the defining minimum in 2.
4. Every two-generated ideal in \(\mathcal O(\mathbb C)\) is principal.
If one generator is zero, the assertion is immediate. Otherwise use <1>2–<1>3. By E-HM213A17-72, since \(F\) and \(G\) have no common zeros, there exist entire \(r,s\) such that \[ Fr+Gs=1. \] Multiplying by \(d\) gives \[ fr+gs=d. \] Thus \(d\in(f,g)\). Conversely \(f=dF\) and \(g=dG\), so both generators belong to \((d)\). Hence \[ (f,g)=(d). \]
5. Every finitely generated ideal is principal.
Proceed by induction on the number of generators. Suppose every ideal generated by at most \(m-1\) functions is principal, and let \[ I=(f_1,\ldots,f_m). \] By the induction hypothesis, \[ (f_1,\ldots,f_{m-1})=(d) \] for some entire function \(d\). Therefore \[ I=(d,f_m). \] By the two-generator case 4, this ideal is principal. Together with 1, the induction proves the claim for every finite number of generators.