Zariski topology of (Krasner) hyperrings
Reza Ameri, Behnam Afshar
TL;DR
The paper investigates the Zariski topology on Krasner hyperrings by analyzing the prime spectrum $Spec(R)$ and establishing criteria for connectivity and irreducibility in terms of hyperring structure such as $R$ not being a nontrivial product and $nil(R)$ being prime. It develops a theory of strongly regular relations and the fundamental relation $\gamma^{*}$, proving a lattice isomorphism with hyperideals containing $\gamma^{*}(0)$ and defining a topology on the set of such relations. It then builds a bridge to classical ring theory by quotienting out by $\gamma^{*}$ to obtain the ring $R/\gamma^{*}$ and constructing a functor between the Zariski topologies on hyperrings and rings, preserving closed sets via $V(J/\gamma^{*})$ and morphisms. Finally, it identifies a topological space $V_{R}$ of prime strongly regular relations, proves its homeomorphism to a subspace of $Spec(R)$, and outlines a program to transfer Zariski topology notions to rings and to applications in sheaves of Krasner hyperrings and hypermodules.
Abstract
In this article, we will study prime spectrum of Krasner hyperrings and Zariski topology on them, which play an important role in algebraic geometry. Then some results about the relationship between the topological properties of Spec(R) and the algebraic properties of the hyperring R will be proved. In the following, by proving that every strongly regular relation on Krasner hyperrings can be considered as a congruence relation, we will define a topology on the set of strongly regular relations, and investigate its relationship with the Zariski topology. In addition, the effect of fundamental relations on the Zariski topology of Krasner hyperrings will also be investigated.