Hyperideal Structure of Krasner’s Induced Quotient Hypperings

Abstract

This paper mainly explores the hyperideal structure of Krasner’s induced quotient hyperrings. By Krasner’s induced hyperring, we mean an additive hyperring R/G induced on a ring R by one of its multiplicative subgroups G. In 1983, Krasner introduced a way of constructing this class of hyperrings and posed the question of whether all the hyperrings that exist naturally were either isomorphic to or could be embedded into this kind of derived hyperrings. Later, in 1985, Massouros proposed a method of construction of hyperfields, which are not embeddable in any of Krasner’s construction. Subsequently, the focus on this important class of additive hyperrings has subsided. We revive the interest in this class of hyperrings by investigating the relationships between the ideals of R and the hyperideals of R/G attentively. Also, we introduce a way of constructing additive hyperrings from given additive hyperrings, and consequently, we prove a few theorems that exhibit the isomorphic relationship between this class of hyperrings under certain conditions.