Syntax for Semantics: Krull’s Maximal Ideal Theorem

Abstract

Krull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations of the Axiom of Choice (AC) in ring theory. For many a consequence of AC, constructive counterparts are well within reach, provided attention is turned to the syntactical underpinning of the problem at hand. This is one of the viewpoints of the revised Hilbert Programme in commutative algebra, which will here be carried out for MIT and several related classical principles.

Towards formal Baer criteria

“…The lying-over principle then asserts that the induced mapping ι * is surjective! Examples include the pull-back of maximal ideals along integral ring extensions [34], as well as the results addressed in the present paper, where flatness is forced by way of totality.…”

“…Our approach requires a slight adaptation of [9,14] which in this section will briefly be outlined. While a first case study has already been carried out in the context of Krull's maximal ideal theorem [34], a thorough and systematic treatment, in particular with regard to inductive generation, will be given elsewhere [38].…”

“…Note further that there is no way around AC proper towards Proposition 3.4. In fact, Proposition 3.4 can be used to derive Krull's maximal ideal theorem [34], and so is in fact ZF-equivalent to AC by way of [21].…”