Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in parti...Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.展开更多
In this paper,we give the equivalent characterizations of principally quasi-Baer modules,and show that any direct summand of a principally quasi-Baer module inherits the property and any finite direct sum of mutually ...In this paper,we give the equivalent characterizations of principally quasi-Baer modules,and show that any direct summand of a principally quasi-Baer module inherits the property and any finite direct sum of mutually subisomorphic principally quasi-Baer modules is also principally quasi-Baer.Moreover,we prove that left principally quasi-Baer rings have Morita invariant property.Connections between Richart modules and principally quasi-Baer modules are investigated.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supported by the Foundation of Natural Science of China(12301029,11171291)Natural Science Fund for Colleges and Universities in Jiangsu Province(11KJB110019 and 15KJB110023).
文摘Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.
基金Foundation item： the National Natural Science Foundation of China （No. 10671122）.
文摘In this paper,we give the equivalent characterizations of principally quasi-Baer modules,and show that any direct summand of a principally quasi-Baer module inherits the property and any finite direct sum of mutually subisomorphic principally quasi-Baer modules is also principally quasi-Baer.Moreover,we prove that left principally quasi-Baer rings have Morita invariant property.Connections between Richart modules and principally quasi-Baer modules are investigated.
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.
