Some Aspects of GH-Rings

In this paper we introduce and study a class of hyperstructure called GH-ring. We show that every commutative group admits a GH-ring structure. Here we obtain necessary and sufficient conditions for a GH-ring to be a division GH-ring (and a strong division GH-ring). The notion of ideals in a GH-ring is also introduced and studied here. We show that every GH-ring with an identity set (i.e., an i-set, in short) always contains a maximal ideal. This is obtained that the maximality of an ideal I of a GH-ring R with condition (R) (i.e., a multiplicative hyperring with absorbing zero) having an i-set, is a necessary and sufficient condition for the quotient GH-ring R/I of R to be a GH-ring. We establish an isomorphism theorem on GH-ring in analogy to the first isomorphism theorem on rings. We construct, over any ring R, a GH-ring structure RA, induced by each A ∈ P* (R) with |A| ≥ 2. We study such GH-ring RA over a ring R, in accordance with the nature of the set A chosen.