HEREDITARY AND COHEREDITARY STRUCTURES - GARMINIA Y. (NIM 30104002), HANNI
This dissertation deals with hereditary and cohereditary properties of algebraic systems such as algebras, modules, coalgebras and comodules. The main contribution of the result is an interrelation of hereditary and cohereditary structures with other properties such as Dedekind and strongly prime and generalization some existing results in literature to the larger classes concerning hereditary and cohereditary properties. The categorical used is applied to develop the results on module and comodule areas. In this dissertation, we investigate on interconection between Dedekind modules and hereditary Noetherian prime (HNP) modules. Particularly, it is shown that any projective Dedekind module is an HNP module. The result is a main contribution of the dissertation in the module theory. It is also shown that the quotient module of a Dedekind module is a Dedekind module. The connection between hereditary modules in o[M] and the projective in o[M] is studied. In this dissertation we investigate the characterizations of a hereditary module connected to a projective module in o[M]. The main result of the dissertation in the comodule theory is about a connection between hereditary coalgebras and hereditary algebras over a commutative self injective ring. Particularly, the dual structure of a cohereditary coalgebra is a hereditary algebra and vice versa. These properties are developed through an auxiliary result; that is any coalgebra being cohereditary if and only if its global dimension being less than or equal to 1. For a coalgebra over a commutative self injective cogenerator ring, it is shown that if its dual is a Dedekind algebra, then the coalgebra is cohereditary strongly coprime. In this dissertation we derive some hereditary and cohereditary properties over a commutative ring; those are the characterization and the direct sum of cohereditary comodule. Further, we apply the results to coalgebras over a commutative ring.