Abstract
A larger class of algebraic hyperstructures satisfying the ring (field)-likeaxioms is the class of Hv-rings (Hv-fields). In this paper, we define the Hv-integraldomain and introduce the Hv-field of fractions of an Hv-integral domain. Also, theHv-quotient ring and some relative theorems are presented. Finally, some interestingresults about the Hv-rings of fractions, Hv-quotient rings and the relations betweenthem are proved.
Highlights
Introduction and preliminaries LetH be a non-empty set and P∗(H) be the non-empty subsets of H
A semi-hypergroup is a hypergroupoid with associative law: (x ∗ y) ∗ z = x ∗ (y ∗ z) for every x, y, z ∈ H; and a hypergroup is a semi-hypergroup with the reproduction axiom: x ∗ H = H ∗ x = H for every x ∈ H
The theory of hyperstructures was introduced by Marty in 1934 during the 8th Congress of the Scandinavian Mathematics [7]. This theory has been studied in the following decades and nowadays by many mathematicians
Summary
Throughout this paper we let R be a commutative hyperring with scalar unit 1 and S is a s.m.c.s. of R It is denoted the operations sum and product of all rings R, S−1R and quotient rings by +, · and we use index for the hyperoperations with the same symbols where is any ambiguity, like ⊕R, ⊕S−1R, +R and +S−1R. The equivalence class containing (A, B) in P∗(R) × P∗(S) is denoted by A, B {[xi, 1], [xi+1, 1]} ⊆ ui, 1 ∈ US and γs∗([r1, 1]) = γs∗([r2, 1]) This Lemma is used in the proof of the following Theorem. For r ∈ ωR and hs in Theorem 2.2, we have γ∗(r) = ωR and γs∗([r, 1]) = hs(γ∗(r)) = hs(ωR) = ωS−1R, because hs is a homomorphism of rings.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
