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.