Principally Left Hereditary and Principally Left Strong Radicals

摘要

A radical γ is normal if and only if γ is principally left hereditary and principally left strong (i.e., γ(L) = L e A and Lz ∈γ for all z ∈ L imply L γ(A)). Let a radical γ satisfy that A°∈γ and S° A° imply S°∈γ.Then γ is a hereditary normal radical if and only if γ is principally left strong and γ {A | (A, +,◇a) ∈γ a ∈ A}, where the multiplication ◇a is defined by x ◇a y ＝ xay. The Behrens radical class B is the largest principally left hereditary subclass of the Brown-McCoy radical class G. Neither3 nor G is principally left strong.