(m,d)-Coherent Rings Relative To Hereditary Torsion Theories And Related Homological Dimensions

In this thesis,we mainly study the generalization of the related theory of(m,d)coherent rings in terms of hereditary torsion theories τ,where m is a positive integer and d is a positive integer or ∞.Two classes of(m,d)-coherent rings relative to the hereditary torsion theory τ,i.e.,τ-(m,d)-coherent rings and L_τ-(m,d)-coherent rings,are investigated.At the same time,a proper subclass of the class of n-flat modules(the class of n-absolutely pure module),i.e.,the class of strongly n-flat modules(the class of strongly n-absolutely pure module),and the strongly n-flat dimension and the strongly n-FP-injective dimension of modules are also studied.Firstly,in order to characterize these two types of generalized(m,d)-coherent rings,two classes of(m,d)-flat modules relative to τ(i.e.,τ-(m,d)-flat modules and L_τ-(m,d)-flat modules)and two classes of(m,d)-injective modules relative to τ(i.e.,τ-(m,d)-injective modules and L_τ-(m,d)-injective modules)are introduced and discussed.As an application,some known characterizations of(m,d)-coherent rings and τ-n-coherent rings can be obtained by corollaries.Then,strongly n-flat modules and strongly n-absolutely pure modules are introduced and their fundamental properties are discussed,which give some new characterizations of n-coherent rings.Finally,the concepts of strongly n-flat dimension and strongly n-FPinjective dimension of modules are introduced,and some properties and characterizations of them are discussed.