Commutative Rings Obtained from Hyperrings (H v -rings) with α∗-Relations

TLDR: The fundamental relation on a hyperring (H v -ring) was introduced by Vougiouklis at the fourth AHA congress as discussed by the authors, which is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring.

Abstract: The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hyperring was introduced by Vougiouklis at the fourth AHA congress. The fundamental relation on a hyperring (H v -ring) is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring. Note that the commutativity with respect to both sum and product in the (fundamental) ring are not assumed. Now, in this article we would like the (fundamental) ring to be commutative with respect to both sum and product, that is, the fundamental ring should be an ordinary commutative ring. Therefore we introduce a new strongly regular equivalence relation on hyperrings (H v -rings). If we consider this relation on a hperring (H v -ring), then the set of quotients is a commutative ring. Some properties of such rings are investigated.