On left (right) strong and left (right) hereditary radicals

Abstract

E.R. Puczylowski proved that the lower radical l M generated by a homomorphically closed class M of rings is left strong if M satisfies the following three conditions: (1) If L ∈ M is a nonzero left ideal of a ring R, then l M (R) = 0, (2) M is hereditary and (3) If I is an ideal of a ring R such that I 2 ≠ 0 and R/I ∈ M, then R ∈ M. We show that the converse of this result does not hold by proving that the lower radical l ∗ ∪ β is left and right strong but the class ∗ ∪ β does not satisfy condition (3), where β denotes the prime radical and ∗ is the class of all semiprime rings R with R/I ∈ β for all nonzero ideals I of R. We prove that the radical l ∗ ∪ β is right and left hereditary although the class ∗ ∪ β is neither left nor right hereditary. This shows that the converse of the well known result, which states that if M is left (right) hereditary then so is l M , does not hold either. We conclude that l ∗∪β is an N-radical. Moreover, we respond to a paper by D.I.C. Mendes in which the author considers a class M of semi-prime rings satisfying condition (*): R ∈ M and L is a left ideal of R imply that L/r(L, L) ∈ M and claims that for such a class the upper radical U(M) determined by M is a left strong radical if and only if U(M) = U(M k ), where r(L, L) = {l ∈ L : Ll = 0} and M k is the essential closure of M. We disprove this claim by showing that the class * satisfies condition (*) and consequently U(∗) is a left strong radical but U(∗ k ) ≠⊂ U(∗).