Auslander-Bridger modules

Classically, the Auslander-Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an arbitrary ring R, which we call Auslander-Bridger modules, with the property that the Auslander-Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander-Bridger right R-modules and isomorphism classes of Auslander-Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander-Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).