The Study Of The Properties Of Hurwitz Series Rings

As is well known, the study of whether some properties of rings and modules are preserved in polynomial rings and modules, or power series rings and modules, or generalized power series rings and modules has become a key words in recent years, and its importance has proved in a number of mathematical literature. In this thesis, we mainly investigate the property of weak Baer, weak quasi-Baer, weak principal quasi-Baer and weak p.p. as well as the associated primes and the weak associated primes of the Hurwitz series rings. It is proved that:(1) If R is a strongly torsion-free Z-module and an NI ring with nil(R) nilpotent, then R is weak Baer, weak quasi-Baer, weak principal quasi-Baer and weak p.p. if and only if the Hurwitz series ring HR is weak Baer, weak quasi-Baer, weak principal quasi-Baer and weak p.p., respectively. (2) Let R be a left perfect ring, MR a right R-module, and Ass(MR) the set of all the associated primes of MR, then we have Ass(HMHR)={HP|P?Ass (MB)}. So the associated primes of the Hurwitz series module HMHR can be described in terms of the associated primes of the right R-module MR. (3) Let R be a right Noetherian semi-commutative ring, NAss(R) the set of all the weak associated primes of R. If R is a strongly torsion-free Z-module, then NAss (HR)={HP|P?NAss (R)}. So the weak associated primes of the Hurwitz series ring HR also can be described in terms of the weak associated primes of the ring R.