Generalized Hopfian modules

Abstract

A module is called generalized Hopfian (gH) if any of its surjective endomorphisms has a small kernel. Such modules are in a sense dual to weakly co-Hopfian modules that were defined and extensively studied in [A. Haghany, M.R. Vedadi, J. Algebra 243 (2001) 765–779]. Several equivalent formulations are given for gH modules and used in their study. Generalized Hopfian modules form a proper subclass of Dedekind finite modules, but this subclass is rather large in that it properly contains Hopfian, and in particular Noetherian modules, as well as Artinian modules and modules with finite dual Goldie dimension. For quasi-projective modules, the properties Dedekind finite, Hopfian and gH are all equivalent, yielding a ring R stably finite iff all finitely generated free R modules are gH. The gH property is shown to be a Morita invariant, and rings are characterized over which all finitely generated modules are gH. Some other classes of rings are also studied. These rings satisfy one of the conditions: all right ideals are gH; all annihilator right ideals are small (right domain); all left regular elements have small right annihilators (right directly finite). We will show: right domain ⇒ right directly finite ⇒ directly finite and these cannot be reversed. These results are by products of the study of special gH modules.