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
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
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 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 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 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.
