Fundamental relation on m-idempotent hyperrings

The γ*-relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R/γ* is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called $\varepsilon^{*}_{m} $, smaller than γ* and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R/ $\varepsilon^{*}_{m} $ is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for Krasner hyperfields, and that $\varepsilon^{*}_{m} $ is a new exhibition for γ* on the above mentioned subclass of m-idempotent hyperrings.