On Chevalley's Extension Theorem

TL;DR

This paper clarifies a correction to a proof in Kaplansky's book, showing that the theorem's statement is still valid as a consequence of Chevalley's theorem despite the identified error.

Contribution

It demonstrates that the corrected statement of Kaplansky's Theorem 56 follows from Chevalley's theorem, resolving the proof error.

Findings

Theorem 56's statement remains valid despite the proof error.

Chevalley's theorem underpins the corrected version of the theorem.

The note clarifies the logical connection between the theorems.

Abstract

Professor Daniel Anderson informed me, recently, that there is an error in the proof of Theorem 56 of Kaplansky's book on Commutative Rings. His (Dan's) reason was "He (Kaplansky) orders by reverse inclusion but in the last line uses inclusion, so we don't contradict maximality (which is minimality)". The aim of this short note is to indicate that while Dan Anderson appears to be correct in pointing out an error in the proof of Theorem 56 of the above mentioned book, the statement of the theorem is a correct consequence of a Theorem of Chevalley.