Comparative analysis of r-hyperideals and r-ideals: From classical algebra to Krasner hyperrings.

In this paper, we explore the concept of r-hyperideals within commutative Krasner hyperrings, extending the classical notion of r-ideals from commutative ring theory into the framework of hyperstructural algebra. Motivated by the structural analogies between rings and hyperrings, we investigate key properties of r-hyperideals and pr-hyperideals, with particular attention to their interaction with regular elements and zero divisors. Furthermore, we study the relationship between prime hyperideals and r-hyperideals, and introduce a relationship with n−hyperideals. Several fundamental results concerning these generalized structures are established. In order to provide a solid foundation for the main results, we first recall basic notions from both classical ideal theory and hyperstructural algebra, setting the stage for a clear comparison between the two frameworks.