The authors develop, continuing on their previous work, an account of Krull’s Maximal Ideal Theorem (MIT) in terms of an infinitary variant of Lorenzen’s and Scott’s entailment relations (relations between finite sets of a given set that are reflexive, monotone and transitive). The discussion is within the framework of Bishop’s constructive mathematics, as formalized in Aczel’s constructive set theory (CZF). After some introductory results on MIT for the case of countable, Noetherian, and arbitrary rings, the authors define geometric entailment relation between finite and arbitrary subsets of a given set, by modifying the conventional entailment relation and then continue on using this notion of entailment. In discussing Krull’s theorem without choice, the authors need inductively generated entailment relation of prime ideals (non-inductive description is through a formal Nullstellensatz). Next they consider the entailment relation of maximal ideals, in order to develop infinitary entailment relations.
In the application section, constructive and elementary characterization of Krull dimension (Kdim) is recalled. One result states that \(\operatorname{Kdim}\mathbf{A}\leq 0\) if and only if \(\vdash_{\text{p}}\,=\,\vdash_{\text{m}}\) (here “p” and “m” stand for prime and maximal ideals respectively). For Jacobson (or Hilbert) rings (every prime ideal is the intersection of all maximal ideals containing it), \(\mathbf{A}\) is a Jacobson ring if and only if \(\vdash_{\text{m}}\) is conservative over \(\triangleright\) and the Jacobson radical is a finitary closure operator. Primary ideals and von Neumann regularity are also discussed and so are integral extensions.
The paper contains almost 100 references related to the subject of the paper.
For the entire collection see [Zbl 1470.03010].