On the Prime Radical of a Hypergroupoid

Abstract

In this study, we give definitions of a prime ideal, a s-semiprime ideal and a w-semiprime ideal for a hypergroupoid K. For an ideal A of K we show that radical of A (R(A)) can be represented as the intersection of all prime ideals of K containing A and we define a strongly A-nilpotent element. For any ideal A of K, we prove that R(A)=∩(s-semiprime ideals of K containing A)= ∩(w-semiprime ideals of K containing A)={strongly A nilpotent elements}. For an ideal B of K put B^(o)=B and B⁽ⁿ⁺¹⁾=(B⁽ⁿ⁾)². If a hypergroupoid K satisfies the ascending chain condition for ideals then (R(A))⁽ⁿ⁾⊆A for some n. For an ideal A of K we give a definition of right radical of A (R₊(A)). If K is associative then R(A)=R₊(A)=R_(A).