In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong （Glasgow Math. J., 48（1）（2006）） negatively. Moreover, some new...In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong （Glasgow Math. J., 48（1）（2006）） negatively. Moreover, some new results about VNL-rings and GVNL-ringsare also given.展开更多
A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left (right) semicentral if for every x ∈ I,xe=exe (ex=exe).And I is called semiabelian if every idempot...A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left (right) semicentral if for every x ∈ I,xe=exe (ex=exe).And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N (I) of nilpotent elements in I is an ideal of I and I /N (I) is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.展开更多
A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring...A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring for every idempotent e of R.We prove that the center of a GVNL ring is also GVNL and every abelian GVNL ring is SGVNL.The formal power series ring R[x] is GVNL if and only if R is a local ring.展开更多
文摘In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong （Glasgow Math. J., 48（1）（2006）） negatively. Moreover, some new results about VNL-rings and GVNL-ringsare also given.
基金The NSF (Y2008A04) of Shandong Province of China
文摘A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left (right) semicentral if for every x ∈ I,xe=exe (ex=exe).And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N (I) of nilpotent elements in I is an ideal of I and I /N (I) is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.
基金supported by the grant of National Natural Science Foundation of China(10971024)the Nanjing University of Posts and Telecommunications(NY209022)
文摘A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring for every idempotent e of R.We prove that the center of a GVNL ring is also GVNL and every abelian GVNL ring is SGVNL.The formal power series ring R[x] is GVNL if and only if R is a local ring.
