Fully dual stable modules

An R-module M is called fully dual stable, if N is contained in Ker f, for every submodule N of M and any homomorphism f of M into M/N. This new concept is dual to full stability and the two concepts are stronger than duo property. Fully dual stable modules stands strictly between duo and multiplication modules. In the class of projective modules, the three concepts coincide. In the class of quasi-projective modules duo and full dual stability coincide. Under certain conditions fully dual stable modules are quasi-projective but not in general. Several properties and characterization have been given for full dual stability. The concept of full dual stability, in fact, depends on submodules and homomorphisms. The researcher considers that some generalizations based on their submodules and others based on their homomorphisms. Maximal and minimal dual stable modules are generalizations of the new concept by modifying the condition on maximal and minimal submodule respectively. Fully pseudo dual stable module is another generalization by modifying the condition on the homomorphism.