On Radicals with Amitsur Property

Abstract

A radical γ has the Amitsur property, if γ(Ax]) = (γ(Ax]) ∩ A)x] for every ring A. To any radical γ with Amitsur property we construct the smallest radical γ ^x which coincides with γ on polynomial rings. Distinct special radicals with Amitsur property are given which coincide on simple rings and on polynomial rings, answering thus a stronger version of M. Ferrero's problem. Radicals γ with Amitsur property are characterized which satisfy Ax, y] ∈ γ whenever Ax] ∈ γ.