Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely...Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.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
This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. The...This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. Then we prove a necessary condition that skew polynomial ring constitutes Armendariz ring. We lastly investigate that condition of skew polynomial ring is a (quasi)-Baer ring, and verify that the conditions is necessary, but not sufficient by example and counterexample.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
Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.
Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a mul...Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a multilinear polynomial f(x_(1),...,x_(n))over C with n non-commuting variables.Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S.Then one of the following statements holds:(a)v∈C and there exist a,b,c∈Q_(r) such that G(x)=ax+bxc for any x∈R with(u+v)a=(u+v)b=0;(b)f(x_(1),...,x_(n))2 is central-valued on R and there exists a∈Q r such that G(x)=ax for all x∈R with ua+av=0.展开更多
基金supported by the National Natural Science Foundation of China(12131015,12071422).
文摘Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.
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.
文摘This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. Then we prove a necessary condition that skew polynomial ring constitutes Armendariz ring. We lastly investigate that condition of skew polynomial ring is a (quasi)-Baer ring, and verify that the conditions is necessary, but not sufficient by example and counterexample.
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.
基金The NSF(1408085QA08)of Anhui Provincialthe Key University Science Research Project(KJ2014A183)of Anhui Province of Chinathe Training Program(2014PY06)of Chuzhou University of China
文摘Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.
基金The work of the second author is partially supported by the National Natural Science Foundation of China(Grant No.10871023).
文摘Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a multilinear polynomial f(x_(1),...,x_(n))over C with n non-commuting variables.Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S.Then one of the following statements holds:(a)v∈C and there exist a,b,c∈Q_(r) such that G(x)=ax+bxc for any x∈R with(u+v)a=(u+v)b=0;(b)f(x_(1),...,x_(n))2 is central-valued on R and there exists a∈Q r such that G(x)=ax for all x∈R with ua+av=0.
基金supported by NSFC(10971024,11201064)the NSF of Jiangsu Province(BK2010393)+1 种基金the NSF of Anhui Educational Committee(KJ2010A126)the Research Culture Funds of Anhui Normal University(2012rcpy040)
