Notes on Brown-McCoy-radicals for Γ-near-rings

Abstract

The Brown-McCoy radical\(\mathcal{G}\) is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals,\(\mathcal{B}\) and\(\mathcal{S}\), respectively, for zerosymmetric Γ-near-rings. Both\(\mathcal{B}\) and\(\mathcal{S}\) are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric Γ-near-ring with left operator near-ringL, it is shown that\(\mathcal{G}\left( L \right)^ + \subseteq \mathcal{B}\left( M \right)\), with equality ifM has a strong left unity.\(\mathcal{G}\) is extended to the variety of arbitrary near-rings, and\(\mathcal{B}\) and\(\mathcal{S}\) are extended to the variety of arbitrary Γ-near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA ⊲N, then\(\mathcal{G}\left( A \right) \subseteq A \cap \mathcal{G}\left( N \right)\), with equality ifA if left invariant.