ON SOME RINGS WHOSE INJECTIVE HULLS HAVE FEW PRERADICAL SUBMODULES

Abstract

A ring R is called strongly prime (DR, CTF, CC) if a(E(R)) = 0 or o(E(R)) = E(R) for all torsion preradicals (idempotent radicals, torsion radicals, cotorsion radicals) σ on Mod-R. (DR and CC rings were introduced recently by Katayama). Examples are provided which distinguish these four conditions one from another and which show each condition to be one-sided. A conjecture of Handelman and Lawrence, to the effect that a ring is CTF if its singular ideal is strongly prime, is disproved, and it is shown that a nonsingular CTF ring is strongly prime iff all of its nonsingular quasi-injective modules are injective. It is also proved that hereditary CTF rings are strongly prime.