Artificial Prime Numbers in Algebraic Structures: A Contextual Extension Beyond the Integers

Abstract

In this paper, the concept of artificial prime numbers, previously defined on subsets of the positive integers, is extended to more general algebraic structures, such as integral domains, Euclidean domains, and rings of algebraic integers. A generalized definition of artificial primality is introduced, based exclusively on the internal divisibility relations of a given set, incorporating the notion of associated elements. Particular cases in the Gaussian integers Z[i] and in non-factorial rings are analyzed, establishing the relationship between artificial primes, irreducible elements, and algebraic primes. Furthermore, fundamental properties, generalized sieving procedures, and constructive methods are discussed, as well as possible applications in pure mathematics, computation, engineering, and STEM education. This approach provides a contextual view of primality, independent of unique factorization, and broadens the conceptual framework of applied number theory.