Some Applications Of Strongly Cotorsion Modules And The Corresponding Dimensions

In this thesis, we study mainly strongly cotorsion modules, the strongly cotorsion dimensions, and their applications. Let R be a ring. A left R-module B is called a strongly cotorsion module if Ext1R(A, B)=0 for any R-module A with finite flat dimension. In chapter 2, we give some equivalent characterizations of strongly cotorsion modules and discuss their basic prosperi-ties. It is proved that a ring with w.gl. dim(R)<∞ if and only if every strongly cotorsion module is injective and l.FFD(R)<∞. It is also shown that (F∞,SC) is a hereditary, complete, perfect cotorsion theory if l.FFD(R)< ∞, where F∞ and SC denote the classes of modules with finite flat dimension and strongly co-torsion, respectively. In chapter 3, we discuss the strongly cotorsion dimension of a module and the global strongly cotorsion dimension of a ring, and we use the global strongly cotorsion dimension of a ring to characterization left perfect rings, left hereditary rings and almost perfect domians. First, we shown that any R-module with finite flat dimension is projective if and only if R is left perfect and l.FFD(R)=0; if and only if any projective module is strongly cotorsion; if and only if l.SCD(R)=0. Secondly, it is also shown that R is left hereditary if and only i l.SCD(R)≤1 and every strongly cotorsion module is injective. And then, for n≥0, we give some new equivalent characterization of left Cn-hereditary rings, i.e, R is left Cn-hereditary if and only if the epimorphic image of strongly cotorsion module is strongly cotorsion; if and only if l.SCD(R)≤1. Finally, we prove that a domain R is an almost perfect domain if and only if l.SCD(R)≤1; if and only if every h-divisible module is strongly cotorsion.