Fundamental commutative (m, n)-rings obtained from (m, n)-hyperrings

Abstract

We introduce the fundamental relation, $alpha^{*}$, on an $(m,n)$-hyperring $R$ and indicate that this relation is the smallest strongly regular equivalence relation on $R$ such that $R/alpha^{*}$ is an commutative $(m, n)$-ring. We use $alpha^{*}$-relation to construct the fundamental functor from category of $(m, n)$-hyperrings to the category of commutative $(m,n)$-rings. Finally, some necessary and sufficient conditions for transitivity of $alpha$ are given.