On uniformly strongly prime gamma rings

Abstract

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(M))mn, where Mmn is the matrix Γnm-ring. τ is shown to be a special radical in the variety of Γ-rings. τ1 is the upper radical determined by the class of usp Γ-rings of bound 1. τ ⊆ τ1, but the reverse inclusion does not hold in general. The place of τ and τ1 in the hierarchy of radicals for Γ-rings is shown.