The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called ri...The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called right exp-DR.We investigate the structures of group rings,right quotient rings,matrix rings and(skew)polynomial rings,through the study of right exp-DR rings.In addition,we provide a method of constructing finite non-abelian p-groups for any prime p.展开更多
It is proved that for matrices A,B in the n by n upper triangular matrix ring T_(n)(R)over a domain R,if AB is nonzero and central in T_(n)(R)then AB=BA.The n by n full matrix rings over right Noetherian domains are a...It is proved that for matrices A,B in the n by n upper triangular matrix ring T_(n)(R)over a domain R,if AB is nonzero and central in T_(n)(R)then AB=BA.The n by n full matrix rings over right Noetherian domains are also shown to have this property.In this article we treat a ring property that is a generalization of this result,and a ring with such a property is said to be weakly reversible-over-center.The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains.The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally.We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.展开更多
This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated...This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.展开更多
文摘The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called right exp-DR.We investigate the structures of group rings,right quotient rings,matrix rings and(skew)polynomial rings,through the study of right exp-DR rings.In addition,we provide a method of constructing finite non-abelian p-groups for any prime p.
基金The second author was supported by the National Natural Science Foundation of China(11361063)the third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education of Korea(2016R1D1A1B03931190).
文摘It is proved that for matrices A,B in the n by n upper triangular matrix ring T_(n)(R)over a domain R,if AB is nonzero and central in T_(n)(R)then AB=BA.The n by n full matrix rings over right Noetherian domains are also shown to have this property.In this article we treat a ring property that is a generalization of this result,and a ring with such a property is said to be weakly reversible-over-center.The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains.The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally.We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.
基金The second author was supported by the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT)(No.2019R1F1A1040405).
文摘This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R)is a commutative regular ring with J(R)nil,where J(R)is the Jacobson radical of R.
