On Orthogonal Dimensions Of Rings

In this thesis, we use functors Ext and Tor to define and investigate orthogonal dimensions if rings and modules. On the one hand, by Ext and Tor orthogonal classes of a general class we investigate Ext and Tor orthogonal dimensions. On the one hand, we use the consistency of⊥(?)-dimension of modules in (?) and (?)⊥-dimension of modules in (?) to investigate the global orthogonal dimension of rings relative to the modules pair ((?),(?)).This thesis consists of four chapters.In Chapter One, we introduce directions and trends on researches related to this thesis, and sum up the groundwork of this thesis.In Chapter Two, for a given class (?) of modules we use the functor Ext to define left Ext-orthogonal dimension (i.e.⊥(?)-dimension), use⊥(?)-resolutions of modules to characterize left Ext-orthogonal dimensions of modules, obtain the relations of left Ext-orthogonal dimensions in the exact sequence, give the case when left Ext-orthogonal dimensions and projective dimensions are equal. Dually, we obtain right Ext-orthogonal dimension i.e.(?)⊥-dimension and dual results. At the same time, we define left and right global orthogonal dimensions of a ring R, obtain some equivalent computing methods of the left global orthogonal dimensions of rings. In particular, for a right semi-Artin ring R we give a simple computing method of the left Ext-orthogonal global dimensions of rings. We also investigate rings with the dimension less than1(i.e. relatively hereditary rings), and give some equivalent characterizations between the left global orthogonal dimensions of rings and the injective envelopes and the⊥(?)-covers with unique mapping property.For a given modules pair ((?),(?)), basing on the former chapter, in Chapter Three we use the consistency of⊥(?)-dimension of modules in (?) and (?)⊥-dimension of mod-ules in (?) to define the global orthogonal dimension ((?),(?))-D(R) of a ring R relative to the modules pair ((?),(?)), give the upper bound of the right global dimension when ((?),(?))-D(R)=0, prove that ((?),(?))-D(R) is less than the upper bound of-(?)-dimensions of all injective modules, characterize when the left and the right orthogonal dimensions of a module and its special preenvelopes and special precovers are equal, consider some properties of the right orthogonal dimensions relative to a special class of modules (i.e. the class FPI of FP-injective modules).In Chapter Four, for a given class (?) of right modules, by the functor Tor we define right Tor-orthogonal dimension of a module(i.e.(?)T-dimension), use (?)T-resolutions of modules to characterize right Tor-orthogonal dimensions of modules, obtain the relations of right Tor-orthogonal dimensions in the exact sequence. We also define right Tor-orthogonal global dimensions of a ring R and obtain some equivalent computing methods.