Right Derived Functors In The Ding Homological Algebra

In1965, S. Eilenberg and J. C. Moore first introduced the viewpoint of relative homological algebra. Since then the relative homological algebra, especially the Gorenstein homological algebra got a rapid development. Nowadays, it has been developed to an advanced level. However, in the most results of Gorenstein homological algebra, the condition’noetherian rings’is essential. In order to make the similar properties of Gorenstein homological algebra hold in a wider environment, Ding Nanqing, Mao Lixin and their coauthors introduced the notions of Gorensteon FP-injective and strongly Gorenstein flat modules. Later on, Gillespie renamed Gorenstein FP-injective modules as Ding injective modules, and strongly Gorenstein flat modules as Ding projective modules. In this paper, we continue to investigate the properties of Ding homological algebra, and the main results obtained are as follows:1. We introduce the notions of (?)-injective,(?)-flat and (?)-projective modules in terms of the right derived functor Ext, and investigate their basic properties. In particular, when (?) is the class of Ding injective modules and (?) is the class of Ding projective modules, the corresponding (?)-injective,(?)-flat and (?)-projective modules are called DI-injective, DI-flat and (?)P-projective modules, respectively. By studying some of the properties of DI-injective, VI-flat and DP-projective modules, we may give some characterizations of classical rings such as semisimple, von Neumann regular, hereditary and semihereditary rings.2. We investigate the Ding right derived functor over arbitrary rings, and study the balance of this functor. Moreover, we give some characterizations of Ding injective and Ding projective dimensions in terms of Ding right derived functors.3. We introduce the notion of strong Tate right derived functor, which connects the usual cohomology functor Ext with the Ding cohomology functor DExt, investigate the balance of this functor and obtain some of the results of Tate cohomology over T2-extension.