A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorph...A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.展开更多
A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper ...A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper （triangular） matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy.展开更多
For a monoid M, we introduce M-McCoy rings, which are generalization of McCoy rings, and we investigate their properties. Every M-Armendariz ring is M-McCoy for any monoid M. We show that R is an M-McCoy ring if and o...For a monoid M, we introduce M-McCoy rings, which are generalization of McCoy rings, and we investigate their properties. Every M-Armendariz ring is M-McCoy for any monoid M. We show that R is an M-McCoy ring if and only if an n × n upper triangular matrix ring αUTn （R） over R is an M-McCoy ring for any monoid M. It is proved that if R is McCoy and R[x] is M-McCoy, then R[M] is McCoy for any monoid M. Moreover, we prove that if R is M-McCoy, then R[M] and R[x] are M-McCoy for a commutative and cancellative monoid M that contains an infinite cyclic submonoid.展开更多
For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is ...For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.展开更多
In this paper,we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions.As a consequence,we generalize several results about this no...In this paper,we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions.As a consequence,we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.展开更多
基金The NSF (10871042,10971024) of Chinathe Specialized Research Fund (200802860024) for the Doctoral Program of Higher Education
文摘A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.
基金The NNSF(10571026)of Chinathe Specialized Research Fund(20060286006)for the Doctoral Program of Higher Education.
文摘A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper （triangular） matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy.
基金the National Natural Science Foundation of China （No. 10171082） the Natural Science Foundation of Gansu Province （No. 3ZSA061-A25-015） and the Scientific Research Fund of Gansu Provincial Education Department （Nos. 06021-21 0410B-09）.
文摘For a monoid M, we introduce M-McCoy rings, which are generalization of McCoy rings, and we investigate their properties. Every M-Armendariz ring is M-McCoy for any monoid M. We show that R is an M-McCoy ring if and only if an n × n upper triangular matrix ring αUTn （R） over R is an M-McCoy ring for any monoid M. It is proved that if R is McCoy and R[x] is M-McCoy, then R[M] is McCoy for any monoid M. Moreover, we prove that if R is M-McCoy, then R[M] and R[x] are M-McCoy for a commutative and cancellative monoid M that contains an infinite cyclic submonoid.
基金Supportd by the Natural Science Foundation of Gansu Province (Grant No. 3ZS061-A25-015)the Scientific Research Fund of Gansu Provincial Education Department (Grant No. 06021-21)
文摘For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.
基金Research is supported by Grant HERMES CODE 30366Departamento de Matemati-cas,Facultad de Ciencias,Universidad Nacional de Colombia,Sede Bogota.
文摘In this paper,we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions.As a consequence,we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.
