Modules Having *-Radical
by
Ayşe Çiğdem Özcan
Department of Mathematics University of Hacettepe, 06532 Beytepe, Ankara-TURKEY

Let R be a ring with identity and M a right R-module. Let E(M) denote the injective hull of M and Z^*(M):=M \cap Rad(E(M)). We call M has *-radical if Z^*(M)=Rad(M). In this note we characterize rings in terms of modules that has *-radical. First we prove that R is a right V-ring (GV-ring) if and only if every (singular) right R-module has *-radical. After that we show that R is a right H-ring if and only if every right R-module that has *-radical is lifting and, R is a semiprimary QF-3 ring if and only if R is right perfect and every projective right R-module that has *-radical is injective (extending). Finally we obtain that R is a QF-ring if and only if every right R-module that has *-radical is projective if and only if Z^*(R)=Jac(R) and every projective right R-module that has *-radical is injective (extending) if and only if Z^*(R)=Z(R) (Z(R) is the singular right ideal of R), R is semiperfect and every projective right R-module that has *-radical is injective (extending).

Date received: November 5, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-08.