Semihyperrings Characterized by Their Hyperideals
M. Shabir, Nayyar Mehmood, Piergiulio Corsini
TL;DR
The paper develops a comprehensive framework for semihyperrings by systematically defining hyperstructures, hyperideals, and related notions, and then characterizing semihyperrings through prime, semiprime, irreducible, and strongly irreducible hyperideals. It establishes fundamental constructions such as hyperideal sums, annihilators, and generated hyperideals, and proves key equivalences, including multiplicative regularity and the behavior of hyperideals under products and intersections. A central contribution is the introduction of the irreducible spectrum topology on the set of irreducible hyperideals $H(R)$, revealing a topological lens on the lattice of hyperideals and linking algebraic and topological properties. The results provide structural tools for analyzing semihyperrings and pave the way for further exploration of their spectra and regularity conditions, with potential applications in algebraic theory and automata-inspired contexts.
Abstract
The concept of hypergroup is generalization of group, first was introduced by Marty [9]. This theory had applications to several domains. Marty had applied them to groups, algebraic functions and rational functions. M. Krasner has studied the notion of hyperring in [11]. G.G Massouros and C.G Massouros defined hyperringoids and apply them in generalization of rings in [10]. They also defined fortified hypergroups as a generalization of divisibility in algebraic structures and use them in Automata and Language theory. T. Vougiouklis has defined the representations and fundamental relations in hyperrings in [14, 15]. R. Ameri, H. Hedayati defined k-hyperideals in semihyperrings in [2]. B. Davvaz has defined some relations in hyperrings and prove Isomorphism theorems in [7]. The aim of this article is to initiate the study of semihyperrings and characterize it with hyperideals. In this article we defined semihyperrings, hyperideals, prime, semiprime, irreducible, strongly irreducible hyperideals, m-systems, p-systems, i-systems and regular semihyperrings. We also shown that the lattice of irreducible hyperideals of semihyperring R admits the structure of topology, which is called irreducible spectrum topology.