We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R...We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.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
Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every id...Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.展开更多
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
We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are rig...We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR.We prove that thepolynomial ring over a ring R is right DR if and only if R is commutative.It is also proved that for a prime number p,the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo,and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.展开更多
The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of re...The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.展开更多
We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we exam...We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible （or semicommutative） rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.展开更多
A ring is said to be right （resp., left） regular-duo if every right （resp., left） regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper tri...A ring is said to be right （resp., left） regular-duo if every right （resp., left） regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of （one-sided） regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.展开更多
文摘We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.
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.
文摘Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.
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.
文摘We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR.We prove that thepolynomial ring over a ring R is right DR if and only if R is commutative.It is also proved that for a prime number p,the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo,and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.
文摘The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.
文摘We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible （or semicommutative） rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.
文摘A ring is said to be right （resp., left） regular-duo if every right （resp., left） regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of （one-sided） regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.
