Annihilator conditions in endomorphism rings of modules

Abstract

The concepts of an intrinsically projective module and an intrinsically injective module are introduced and their connection with the presence of annihilator conditions in the endomorphism ring of a module is explained. It is shown that a ring R is quasi-Frobenius if and only if in the endomorphism ring of any fully projective (or any fully injective) R-module it is true that r(l(I))=I andl(r(J))=J for all finitely generated right ideals I and finitely generated left ideals J.