Prime ideals of strong semigroup graded rings have been characterized by Bell, Stalder and Teply for some important classes of semigroups. The prime ideals correspond to certain families of ideals of the component rings called prime families. In this paper, we define the notion of a primitive family and show that primitive ideals of such rings correspond to primitive families of ideals of the component rings.
Gopalakrishnan, Hema, "Primitive Ideals of Semigroup Graded Rings" (2004). Mathematics Faculty Publications. 19.