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

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

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

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\u2019s 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\u2019s 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\u2019s 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

Codification, Abstraction, and Firm Differences: A Cognitive Information-based Perspective

knowledge, codification, theory of the firm, firm heterogeneity, resource-based view, information asymmetry, B52, D01, D82, D83, L25,