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
Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions:(1)if for each x∈R\N(R)and each y∈R,(xy)^(k)=x^(k)y^(k)for k=m,m+1...Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions:(1)if for each x∈R\N(R)and each y∈R,(xy)^(k)=x^(k)y^(k)for k=m,m+1,n,n+1,where m and n are relatively prime positive integers,then R is commutative;(2)if for each x∈R\J(R)and each y∈R,(xy)^(k)=y^(k)x^(k)for k=m,m+1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.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This work was in part supported by the National Science Foundation of China under Grant Nos.11701499 and 11671008the National Science Foundation of Projects of Jiangsu Province of China under Grant No.BK20170589.
文摘Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions:(1)if for each x∈R\N(R)and each y∈R,(xy)^(k)=x^(k)y^(k)for k=m,m+1,n,n+1,where m and n are relatively prime positive integers,then R is commutative;(2)if for each x∈R\J(R)and each y∈R,(xy)^(k)=y^(k)x^(k)for k=m,m+1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.
