Modules whose Injective Endomorphisms Are Essential

Abstract

An R-module M is called weakly co-Hopfian if any injective endomorphism of M is essential. The class of weakly co-Hopfian modules lies properly between the class of co-Hopfian and the class of Dedekind finite modules. Several equivalent conditions are given for a module to be weakly co-Hopfian. Being co-Hopfian, weakly co-Hopfian, or Dedekind finite are all equivalent conditions on quasi-injective modules. Some other properties of weakly co-Hopfian modules are also obtained. The ring R is said to be right strong stably finite if all the finitely generated free right R-modules are weakly co-Hopfian. We shall characterize such rings and show that they are stably finite and satisfy the right strong rank condition. Examples show that stably finite rings and rings with the right strong rank conditions need not be strong stably finite. Both weakly co-Hopfian and right strong stably finite are Morita invariants, although the right and left strong stably finite are different properties. The class of commutative rings and the class of rings with finite right uniform dimension are proper subclasses of the class of right strong stably finite rings. We shall also investigate conditions that are relevant to weakly co-Hopfian modules. Equivalent statements are found on a ring to have all its finitely generated right modules weakly co-Hopfian.