Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]]...Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,（R） has a generalized join in S,（R）. Several known results follow as consequences of our results.展开更多
For a ring endomorphism α, in this paper we introduce the notion of s-power- serieswise nil-Armendariz rings, which are a generalization of α-power-serieswise Armendariz rings. A number of properties of this general...For a ring endomorphism α, in this paper we introduce the notion of s-power- serieswise nil-Armendariz rings, which are a generalization of α-power-serieswise Armendariz rings. A number of properties of this generalization are established, and the extensions of α- power-serieswise nil-Armendariz rings are investigated. Which generalizes the corresponding results of nil-Armendariz rings and power-serieswise nil-Armendariz rings.展开更多
Let R be a ring such that all left semicentral idempotents axe central and α a weakly rigid endomorphism of R. It is shown that the skew power series ring R[[x; α]] is right p.q.Baer if and only if R is right p.q.Ba...Let R be a ring such that all left semicentral idempotents axe central and α a weakly rigid endomorphism of R. It is shown that the skew power series ring R[[x; α]] is right p.q.Baer if and only if R is right p.q.Baer and any countable family of idempotents in R has a generalized join in I(R), where I(R) is the set of all idempotents of R.展开更多
A ring R is called a pseudo weakly clean ring i[ every element xE R can be written in the form of x=e＋u＋（1-e）rx or x=-e＋uq-（1-e）rx where e is an idempotent and u is a invertible element. These ringsare shown to...A ring R is called a pseudo weakly clean ring i[ every element xE R can be written in the form of x=e＋u＋（1-e）rx or x=-e＋uq-（1-e）rx where e is an idempotent and u is a invertible element. These ringsare shown to be a unifying generalization of skew power series ring R[[x;σ]], Hurwitz series ring H（R） andT（R,a）. The pseudo weak cleanness of the ring o[ triangular matrices is discussed as well. Furthermore, thispaper proves that the following are equivalent： that is R is pseudo weakly clean; there is an integer n such thatR[x]/（x^n） is pseudo weakly clean; there is an integer n such that R[[x]]/（x^n） is pseudo weakly clean.展开更多
Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition unde...Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.展开更多
基金The Youth Foundation(QN2012-14)of Hexi University
文摘Let R be a ring and （S, 〈） be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,（R） has a generalized join in S,（R）. Several known results follow as consequences of our results.
文摘For a ring endomorphism α, in this paper we introduce the notion of s-power- serieswise nil-Armendariz rings, which are a generalization of α-power-serieswise Armendariz rings. A number of properties of this generalization are established, and the extensions of α- power-serieswise nil-Armendariz rings are investigated. Which generalizes the corresponding results of nil-Armendariz rings and power-serieswise nil-Armendariz rings.
基金National Natural Science Foundation of China (10171082), TRAPOYT the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China
文摘Let R be a ring such that all left semicentral idempotents axe central and α a weakly rigid endomorphism of R. It is shown that the skew power series ring R[[x; α]] is right p.q.Baer if and only if R is right p.q.Baer and any countable family of idempotents in R has a generalized join in I(R), where I(R) is the set of all idempotents of R.
文摘A ring R is called a pseudo weakly clean ring i[ every element xE R can be written in the form of x=e＋u＋（1-e）rx or x=-e＋uq-（1-e）rx where e is an idempotent and u is a invertible element. These ringsare shown to be a unifying generalization of skew power series ring R[[x;σ]], Hurwitz series ring H（R） andT（R,a）. The pseudo weak cleanness of the ring o[ triangular matrices is discussed as well. Furthermore, thispaper proves that the following are equivalent： that is R is pseudo weakly clean; there is an integer n such thatR[x]/（x^n） is pseudo weakly clean; there is an integer n such that R[[x]]/（x^n） is pseudo weakly clean.
基金Supported by National Natural Science Foundation of China (Grant No.10961021)the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China
文摘Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.
基金Supportedin part by the Natural Science Foundation of Zhejiang Province(Y6090404)supported in part by the Graduate Innovation Seed Project of Hangzhou Normal University
