The purpose of this article is to present certain results arising from a study of theory of hyperrings. By a hyperring we mean a Krasner hyperring, that is, a triple (R, +,·) is such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup with a zero 0 where 0 is the scalar identity of (R, +) and · is distributive over + . In this article, we define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions we obtain some results. We define hyperring of fractions and hyper-valuation on a hyperring. For this, as in the classical case, we need a mapping from R onto an ordered group G. Finally, we shall state and prove the Chinese Remainder Theorem for the case of hyperrings.

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.

In this article, we introduce the notion of quasi-hyperideal in semihypergroups, and moreover, we introduce the notions of an (m, n)-quasi-hyper-ideal, n-right hyperideal, and m-left hyperideal in semihypergroups, and relations between them are studied. We obtain different characterizations concerning different properties of (m, n)-quasi-hyperideals, minimal (m, n)-quasi-hyperideals, minimal m-left hyperideals, and minimal n-right hyperideals, and relations between them are investigated. Also, some intersection properties and characterizations of (m, n)-quasi-hyperideals of semihyperideals and regular semihyperideals have been studied. In the sequel, m-left simple, n-right simple, and (m, n)-quasi-simple semihypergroups are defined, and some properties of them are investigated.

One of the aims of this article is to extract, whenever possible, the common elements of several seemingly different types of algebraic hypersystems. In achieving this, one discovers general concepts, constructions, and results which not only generalize and unify the known special situations, thus leading to an economy of presentation, but, being at a higher level of abstraction, can also be applied to entirely new situations, yielding significant information and giving rise to new directions. We shall consider a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups, and n-ary semigroups. This new class of hypersystems is called n-ary hypersemigroups and properties of such hypersemigroups are investigated. On an n-ary hypersemigroup, we describe the smallest equivalence relation β* whose quotient is an ordinary n-ary semigroup.