On Prime and semiprime ideals of $Gamma$-semihyperrings

The $Gamma$-semihyperring is  a generalization of the concepts of a semiring, a semihyperring and a $Gamma$-semiring. In this paper, the notions of completely prime ideals and prime radicals for $Gamma$-semihyperring are introduced and studied some important properties accordingly. We also introduced the notions of $m$-system and complete $m$-system. Then characterizations of prime ideals and completely prime ideals of $Gamma$-semihyperring with the help of $m$-system and complete $m$-system has been taken into account. It is our attempt to find a bridge between semiprime (completely semiprime) ideals and prime (completely prime) ideals of a $Gamma$-semihyperring.