Remarks on a remark of Kaplansky

Abstract:

In his book Commutative rings, Kaplansky makes an interesting remark following the proof of the Hilbert Basis Theorem. He says, "Justly celebrated though this proof is, it leaves one somewhat dissatisfied, since the condition that $I$ and the ${I_n}$ ’s be finitely generated is by no means necessary for $J$ to be finitely generated". The purpose of our note is to elaborate on Kaplansky’s remark.