ENDOMORPHISM RINGS OF FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN - Romaldy Stephanus, Johannes
This thesis deals with an identication of endomorphism rings of nitely generated modules over a principal ideal domain with matrix rings. A fact that will be used is that a nitely generated module over a principal ideal domain can be decomposed into a direct sum of its torsion submodule and a free submodule. Furthermore, the torsion submodule can be decomposed into a direct sum of primary submodules and each primary submodules can be decomposed into a direct sum of cyclic submodules. On the other hand, the free submodule can be decomposed into a direct sum of cyclic submodules generated by elements in its basis. In this thesis, it is shown that the endomorphism rings of nitely generated modules over a principal ideal domain can be identied by a 2 2 upper block matrix ring where block-(11) represents the endomorphism ring of torsion submodule, block-(12) represents the homomorphism module from the free submodule to the torsion submodule, and block- (22) represents the endomorphism ring of free submodule. Details of each block is also presented in this thesis.