Zariski Topology of (Krasner) Hyperrings

Abstract

In this paper, we investigate the Zariski topology on the prime spectrum of commutative Krasner hyperrings and explore its interplay with the underlying algebraic structure. We characterize the topological properties of the spectrum, such as connectedness, irreducibility, compactness, and separation axioms, and provide necessary and sufficient conditions for each. Notably, we show that the spectrum is irreducible if and only if the nilradical is a prime hyperideal, and it is connected precisely when the hyperring is not a non-trivial product. We also study functorial behavior of the Zariski topology in the category of hyperrings and analyze its correspondence with classical ring theory via the fundamental relation γ∗.  Furthermore, we define a topology on the space of prime strongly regular relations and establish a homeomorphism with a subspace of the classical spectrum. These results contribute to the categorical and topological foundations necessary for developing a sheaf-theoretic framework in the context of hyperrings.