Based on the concept of accessible subhemirings and inspired by the work on the general
Kurosh-Amitsur radical theory for rings, this paper studies the lower radical classes and the
hereditary radical classes of hemirings. We characterize radical classes of hemirings, and con-
struct a lower radical class from a homomorphically closed class. We provide a necessary and
sufficient condition under which an upper radical class of hemirings becomes hereditary and
prove that an upper radical class of a regular class of semirings is hereditary. Besides, we show
that the Brown-McCoy radical class and a Jacobson-type radical class are hereditary.