Abstract

We develop an “external” Kurosh–Amitsur radical theory of semirings and obtain some fundamental results regarding the Jacobson and Brown–McCoy radicals of hemirings. Among others, we single out the following central results: characterizations and descriptions of semisimple hemirings; semiring versions of the classical Nakayama's and Hopkins's Lemmas and Jacobson–Chevalley Density Theorem; the fundamental relationship between the radicals of hemirings R and matrix hemirings M n (R); the matric-extensibleness (see, e.g., [4, Section 4.9]) of the radical classes of hemirings; the Morita invariance of the Jacobson– and Brown–McCoy-semisimplicity of semirings.