The Study Of The Homlogical Properties Of The Hurwitz Series Rings

Firstly,in this thesis we study the relationship between the flatness,or the faithful flatness of the Hurwitz series residue ring HR/(f)or the Hurwitz polyno mial residue ring hR/(f)and the f,which is the generated element of the principal ideal(f).Under the conditions that R is a perfect coherent strongly torsion free commutative ring and f? HR or f ? hR it is proved that Hurwitz series residue ring HR/(f)is a flat R-module whenever there exists some positive integer n ? N such thatf(k)?Id(R)for all 0?k?n-1 and f(n)? U(R);The Hurwitz polyno mial residue ring hR/(f)is a flat R-module wheneverf(0)?U(R)and the Hurwitz polynomial residue ring hR/(f)is a faithful flat R-module wheneverf(?(f))?Id(R)and f(?(f)-1)?U(R).Secondly,we investigate whether the strongly Hopfian property or the acc on d-annihilators property of the ring R is preserved under the Hurwitz series extension or the Hurwitz polynomial extension.It is proved that if the ringi R is a torsion freeZ-module,then the ring R is a strongly Hopfian ring if and only if the Hurwitz polynomial ring hRis a strongly Hopfian ring.Moreover,it is also proved that if R is a reduced commutative ring and a torsion free Z-module,then R satisfies acc on d-annihilatorsif and only if hR satisfies acc on d-annihilators.Finally,we describe some radicals of Hurwitz series rings.It is proved that if the lower nilradical nil*(R)and the upper nilradical nil*(R)is nilpotent respectively,then the corresponding lower nilradical nil*(HR)and the corresponding upper nilradical nil*(HR)is also nilpotent respectively and nil*(HR)=H(nil*(R)),nil*(HR)=H(nil*(R)).If the ring R is a strongly torsion free NI ring and nil(R)is nilpotent,then nil(HR)is nilpotent and nil(HR)=H(nil(R)).If the ring Ris a strongly torsion free NI ring and nil(R)is nilpotent,then we have thet nil*(HR)=nil*(HR)=nil(HR)=H[nil*(R))=H(nil*(R))=H(nil(R)).