A Realization of Hyperrings

Abstract

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.