Strongly Torsion-free Dimensions Of Modules And Global Strongly Torsion-free Dimensions Of Rings

Let R be a ring and D a right R-module. If Tor1R (D, M)= 0 for all left R-modules M with finite flat dimension, then D is called a strongly torsion-free. In this thesis, our main goal is to characterise strongly torsion-free modules and to investigate strongly torsion-free dimensions of modules. In chap-ter 2, we give some equivalent characterizations of strongly torsion-free modules and discuss their basic prosperities. It is shown that (D∞,F∞) is a Tor-torsion theory if and only if l.FFD(R)<∞, where D∞ and F∞ denote the classes of strongly torsion-free right R-modules and left R-modules with finite flat dimen-sion, respectively. Then it is also shown that every right R-module is strong-ly torsion-free if and only if l.FFD(R)= 0. In chapter 3, we discuss strongly torsion-free dimensions of modules and global strongly torsion-free dimensions of rings. We prove that if l.FFD(R)<∞, then l.FFD(R)= r.stf.dim(R), where r.st f. dim(R) is the global strongly torsion-free dimension of ring R. In chapter 4, we define st-VN regular rings and prove that R. is a right st-VN regular ring if and only if l.FFD(R)= 0; if and only if every left R-module with finite flat dimension is flat:if and only if every finitely presented right R-module is strongly torsion-free. Besides, we introduce the concept of STH rings in this thesis. It is proved that R is a right STH ring and w.gl. dim(R)< oo if and only if R is a right STH ring and every strongly torsion-free module is flat; if and only if iv.gl. dim(R)≤1.