# An algorithmic approach to the existence of ideal objects in commutative algebra

## TL;DR

This paper introduces a computational approach to ideal objects in commutative algebra, using sequential algorithms and interpretative methods to replace non-constructive existence proofs with constructive procedures.

## Contribution

It provides a novel algorithmic interpretation of maximality principles and demonstrates their application in understanding prime ideals and the nilradical in commutative rings.

## Key findings

- Developed a state-based algorithm for maximality principles.
- Provided an algorithmic proof that the intersection of prime ideals is in the nilradical.
- Bridged classical existence results with constructive, computational methods.

## 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.