Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition Original Research Article

Let P be a quasi-projective module and let E be its endomorphism ring. The condition that E has one in the stable range is shown to be equivalent to a number of conditions involving mappings of modules, and completions of diagrams by automorphisms. The dual problems for quasi-injective modules are also considered. The condition that principal right ideals of E are uniquely generated is also characterized by diagram completion properties. As consequences of these results, several sufficient conditions are given for the equivalence of a pair of endomorphisms of P.