A note on minimal generators of hereditary radicals

Abstract

In a recent paper [1] we defined, for a ring R, the class T~EO~M 1 ([1], Theorem 4). An HH class M is a minimal generator for the radical P = LM if and only if F(R) fq M ~ 0 for all R ~ M, and TH~OI~M 2 ([1], Theorem 6). For a non-zero HH class M, the radical P ~-LM has a minimal generator if and only if every 0 :/= R E M has an ideal 0 ~ I which is either nilpotent or is a ring for which F(I) f3 M ~ O.The following result contains all known cases of hereditary radicals with minimal generators: THEOREM 3. Let M be an HH class. Then P ~--LM has a minimal generator if/or each R E M either: