| Let R be an associative ring with a unit and n an integer greater than 0.This paper is mainly composed of the following three parts:In the first part,we study the n-phantom morphisms and n-Ext-phantom morphisms in the category of left R-modules,and by this we characterize the flat dimension of modules and FP-injective dimension of modules,respectively.In the second part,we consider the cokernel of n-phantom envelope which is monic and the kernel of n-Ext-phantom cover which is epic,and then,we characterize the weak dimension of a ring using n-Ext-phantom cover.The third part,we first discuss the relationship between n-phantom morphisms and Torn-monomorphisms(epimorphisms)and between n-Ext-phantom morphisms and Extn-monomorphisms(epimorphisms)in the following commutative diagram with exact rows.(?) And then,we exhibit a transitivity of Torn-monomorphisms and Extn-epimorphisms on its degree n.Finally,we provide some equivalent characterizations of a morphism f:M→ N which is a Torn-monomorphism,where both M and N are left R-modules. |
