Abstract

The purpose of this paper is to extend the results given by J.S. Golan [2] on Gel’fand additively-idempotent and additively-regular semirings, and the results obtained by N.X. Tuyen and his coauthors [3–5] on semimodules. In particular, by generalizing the results developed by F.W. Anderson and K.R. Fuller [1] on radicals of rings and modules. We consider the radicals of semirings and prove some of their properties (Theorems 5, 7 and 16). We show that radicals of semirings can be described by quasiregular elements of semirings (Propositions 14 and 15). In particular, we give the modification of Nakazama’s lemma for semirings (Corollary 18).