On S-versions of chain conditions over modules

The aim of this study is to define and introduce various generalizations of ascending and descending chain conditions. Using these generalizations, the thesis provides a wide range of characterizations for many rings and modules. In Section 2, the concept of strongly dccr^⋆ modules is introduced and studied. Strongly dccr^⋆ condition generalizes the class of Artinian modules and it is stronger than dccr^⋆ condition. The concept is fully characterized in terms of perfect rings and in terms of principally cogenerated modules. Also, the relationships between the concept and strongly special modules are examined. Moreover, versions of Union Theorem and Nakayama's Lemma are given in light of strongly dccr^⋆ concept. In Section 3, the thesis defines the class of S-Artinian modules, where S is a multiplicatively closed set. S-Artinian notion is broader than the notion of Artinian modules. Furthermore, the concept of finitely S-cogenerated modules is defined. The relationship between these two concepts is studied, and many characterizations of S-Artinian are given in terms of several modules and rings. In Section 4, S-Noetherian Spectrum condition is introduced. Many properties and characterizations of S-Noetherian spectrum condition are examined. Moreover, an analogous result to Cohen's theorem for modules satisfying S-Noetherian spectrum condition is proven. Furthermore, modules having Noetherian spectrum are characterized in terms of modules satisfying the S-Noetherian spectrum condition. In the last part of the thesis, the class of modules satisfying S-dccr (S-dccr^⋆) condition is introduced. S-dccr (S-dccr^⋆) condition is a generalization of S-Artinian modules. This section extends many properties and characterizations of Artinian, S-Artinian and dccr modules to S-dccr condition. Furthermore, S-versions of Union Theorem and Nakayama's Lemma are given in light of S-dccr concept.