The Dimensions Of Rings And Modules

In this thesis, we will discuss the homological dimension and homological properities of rings and modules which is one of the most impertinent parts in homological theory.In the first chapter,we investigate the relation of homological properities between ring R.[[X]] and ring R, we prove that a module M is an injective or a flat R-module,then it is an injective or a flat R[[X]]-module respectively,and if M is an injective or a flat R-module,then Hom(R[[X]], M) is an injective R[[X)]-module,R.[[X}] M is a flat R{[X]}-module. In Theorem 3 of this chapter,we prove that Homological properities and homological dimensions of residue ring play very important roles.The homological properities and homological dimension of coordinate ring have been studied in the thesis of doctor Yang Jing-hua.In the second chapter,we generalize the results of coordinate ring to power se-ries residue ring. Under some conditions ofthe coefficient of f(x), we investigate the fathful flatness and other related properities of ring R[[X]]/(f(x)) as R-module and obtain it's global dimension. Futhermore,let be a power series in the ring if satisfies some conditions,we also obtain the global dimension of ring .The mainly result of this chapter is that if GD(R[[X]]/ (f(x)), GD(R) are finites,thenGp(R([X]]/(f(x))) # GD(R)#GD(R[[X]]/(f(x)))+pdR(R{[X]}/(f(x))).Given a ring,one always takes the supremum of some homological dimension of specified module to obtain the corresponding global dimension,and to characterize (he ring. In the third chapter,we introduce the concept of homo-logical dimension of M- projective module and M-left global dimension of a ring,obtain the relation between M-left global dimension and left dimension as well as some equivalent statements of M- global dimension 0Given a ring or module ,one can define various homological dimensions by resolving the module.In the fourth chapter,we introduce the concept of M-extension functor and apply MExt to invesigate M-injective dimension of module and M- left global dimension of a ring.Using the theories of isomorphic functors and isomorphic equivalent categories to show that R and Mn(R) have equal left global dimension is complicated, In the fifth chapter,we obtain the global dimension of a kind of MoritaContext by using the technique of martix calculation and torsion func-tor,furthermore we prove tha,tLgdMn(R) ?LgdR in a very easy way .