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On a general hyperring, there is a fundamental relation, denoted  γ  * , such that the quotient set is a classical ring. In a previous paper, the authors defined the relation  ε m   on general hyperrings, proving that its transitive closureε   m    ∗$\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$    is a strongly regular equivalence relation smaller than the  γ  * -relation on some classes of hyperrings, such that the associated quotient structure moduloε   m    ∗$\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$    is an ordinary ring. Thus, on such hyperrings,ε   m    ∗$\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$    is a fundamental relation. In this paper, we discuss the transitivity conditions of the  ε m  -relation on hyperrings and  m -idempotent hyperrings.

Transitivity of the ε m -relation on ( m -idempotent) hyperrings