Direct Limits of Krasner Hyperrings and their Hyperideals

Direct limits provide a fundamental tool for constructing algebraic structures over directed systems. In this paper, we investigate direct limits in the setting of Krasner hyperrings. After reviewing the necessary background on hypergroups and Krasner hyperrings, we introduce the notion of a direct system of Krasner hyperrings and establish the existence of its direct limit. We prove that the direct limit of such a system is again a Krasner hyperring. Furthermore, we study the behavior of hyperideals under this construction, showing that compatible families of hyperideals induce a hyperideal in the direct limit, and conversely, every hyperideal of the limit arises in this way. Finally, we construct the quotient of a direct limit and establish its compatibility with the direct limit construction. These results extend classical properties of direct limits to the framework of hyperring theory.