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.

Using a new definition, with respect to [21], for normal hyperideals in Krasner ( m, n )-hyperrings, we show that the corresponding quotient structures are ( m, n )-rings. We prove equivalency of (strongly) regular and (strongly) compatible relations on n -ary hypergroups. Also, the connection between ( m, n )-hyperfields and maximal normal hyperideals is investigated.

The aim of this study is to extend the concept of normality of a subfuzzy hypermodule to subfuzzy (m, n)-hypermodules. Hence, normal subfuzzy (m, n)-hypermodules of a fuzzy (m, n)-hypermodule, denoted by NFS_mnHM, is introduced and studied. In this way, normal subfuzzy (m, n)-hypermodules are used to construct quotient fuzzy (m, n)-hypermodules endowed with fuzzy n-ary hyperoperations to obtain some basic results in this context.