Prime Spectrum and Relations on Krasner Hypermodules

We study the Zariski topology of Krasner hypermodules from the perspective of regular relations. To this end, we first examine the main properties of the Zariski topology of hypermodules and investigate its connection with the classical case. In the next step, we prove results regarding the relationship between relations on the R-hypermodule M , and relations on the hypergroup M and the hyperring R. We then show that, (strongly) regular relations on a hypermodule and specific subhypermodules form isomorphic lattices. This results will also be examined for quotient hypermodules. Next, we define prime and primary relations on hypermodules and examine their properties. These properties help us introduce a topology based on relations, which plays an important role in the study of sheaves of hypermodules.