Factorization of Ideals in Commutative Domains and Some Generalizations of Dedekind Domains

Abstract

Dedekind domain are the rings in which every nonzero proper ideal has a prime factorization. In this thesis, we first study valuation rings and Prüfer domains. Then we characterize Dedekind domains which are also Noetherian Prüfer domains. After studying the characterizations of Dedekind domains and the prime factorization, we give some generalizations of Dedekind domains such as almost Dedekind domains and SP-domains. Almost Dedekind domains are ring with all localizations are DVRs. We study cancellation law for ideals to give a characterization of almost Dedekind domains. Then we study SP-domains, in which every ideal has a radical factorization, which can be uniquely determined under some conditions. After giving the fact that SP-domains are almost Dedekind domains, we study the correspondence between lattice ordered abelian groups and Bezout domains. In the last section, we give two examples of almost Dedekind domains.