Auslander-Bridger transpozu üzerine

Let R be a ring with unity. A module M is said to be stable if it has no nonzero projective direct summand. It is well-known that over a semiperfect ring R, taking the Auslander-Bridger transpose establishes a one to one correspondence between the isomorphism classes of finitely presented stable right R-modules and the isomorphism classes of finitely presented stable left R-modules. Over a general ring R, Facchini and Girardi extend this one to one correspondence by taking an Auslander- Bridger transpose between two isomorphism classes of some modules which they call Auslander-Bridger right R-modules and Auslander-Bridger left R-modules. In this thesis, we give some ring classes over which every finitely presented module can be decomposed into a direct sum of a projective submodule and a stable submodule. If the ring R is semilocal has finite hollow dimension as a right R-module or left semihereditary, then every finitely presented right R-module has such a decomposition. We give some ring examples over which there exists a finitely generated module or a finitely presented module where such a decomposition fails. Another problem that we deal with in this thesis is the Auslander-Bridger transpose of finitely presented simple modules. It is shown by Türkoğlu in his master thesis that if R is a commutative ring where every maximal ideal is finitely generated and projective, then an Auslander- Bridger transpose of every simple R-module is isomorphic to itself, and so simple. A related result is given by Haack. We give the details of the proof of Haack's result: for an Artinian ring R, the Auslander-Bridger transpose of every nonprojective simple right R-module is simple if and only if R is a serial ring. This also gives us some relations between the proper classes of short exact sequences of R-modules that are projectively generated or flatly generated by simple modules.