The Homological Properties And The Homological Dimensions Of The Residue Rings

Let R be an associative ring with identity,? an automorphism of the ring R,? an?-derivation of the ring R,R[x;?,?]and R[[x;?]]the Ore extension ring and the skew power series ring over R,respectively,(?) the skew polynomial of the Ore extension ring R[x;?,?]and f(x)=?i=0naixi the skew power series of the skew power series ring R[[x;?]].Firstly,in this thesis,it is investigated the relationships between the flat or faithfully flat property of the Ore extension residue ring R[x;?,?]/(f(x))and the coefficients of the skew polynomial f(x),and it is showed that if some coefficients of the skew polynomial f(x)satisfies some additional conditions,then the Ore extension residue ring R[x;?,?]/(f(x))is a flat or a faithfully flat R-module;Also it is studied the relationship between the left global dimension of the Ore extension residue ring R[x;?,?]]/(f(x))and that of the base ring R,and it is proved that if some coefficients of the skew polynomial f(x)satisfies some conditions,then the left global dimension of R[x;?,?]/(f(x))can be described in terms of the left global dimension of the base ring R.Secondly,it is explored the relationships between the flat or faithfully flat property of the skew power series residue ring R[[x;?]]/(f(x))and the coefficients of the skew power series f(x),and it is showed that if some coefficients of the skew power series f(x)satisfies some additional conditions,then the skew power series residue ring R[[x;?]]/(f(x))is a flat or a faithfully flat R-module;Moreover,it is showed that the global dimension of R[[x;?]]/(f(x))can be characterized in terms of the global dimension of the base ring R whenever some coefficients of the skew power series f(x)satisfies some conditions.Consequently,the theories of the homological property or the homological dimension relating to the Ore extension residue ring R[x;?,?]/(f(x))or the skew power series residue ring R[[x;?]]/(f(x))are extended to a more general setting.