An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra

Abstract

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

A universal algorithm for Krull's theorem

“…A simple and well-known consequence of this theorem is the following, which was already analysed in [36] and will be examined in more generality in Section 5.…”

“…We present several examples in the second part of the paper. Our paper extends some initial ideas in this direction presented in [36]: there, we focus on a very simple instance of Krull's theorem, whereas our more abstract framework here not only allows us to consider a range of different applications, but could be readily applied to other case studies in future work.…”

On the computational content of Zorn's lemma

“…Moreover, given some countable ring R whose elements can be coded up as natural numbers and whose operations + R and • R represented as primitive recursive functions N×N → N, the existence of a maximal ideal in R would be provable in PA ω + ZL [],⊕,≺ . We do not give full details of this (an outline of the formalisation can be found in [21]). Instead we simply sketch why both S ω and C ω are satisfy ZL [],⊕,≺ and are thus models of PA ω + ZL [],⊕,≺ .…”

Bounds for Indexes of Nilpotency in Commutative Ring Theory: A Proof Mining Approach

“…The classical theorem and the application given in §4 were very recently the subject of attention of the paper [16]. The focus of this paper is, nevertheless, different from ours: It is on maximal ideals (instead of prime ideals) and on precise computational witnesses (instead of bounds) and, in order to calculate these witnesses, some computational assumptions are made.…”