Let R be a prime ring of characteristic different from 2,Q be its maximal right ring of quotients,and C be its extended centroid.Suppose that f(x1,...,xn)is a non-central multilinear polynomial over C,0≠p∈R,and F,G ...Let R be a prime ring of characteristic different from 2,Q be its maximal right ring of quotients,and C be its extended centroid.Suppose that f(x1,...,xn)is a non-central multilinear polynomial over C,0≠p∈R,and F,G are two b-generalized derivations of R.In this paper we describe all possible forms of F and G in the case pG(F(f(r))f(r))=0 for all r=(r1,...,rn)in R^n.展开更多
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
Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t≥> 1 fixed integers with t ≤ m + n + s. Suppose that a is a non-trivial automorphism of R and let φ(x,...Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t≥> 1 fixed integers with t ≤ m + n + s. Suppose that a is a non-trivial automorphism of R and let φ(x,y)=[x,y]^t -[x,y]^m[α([x,y]),[x,y]^n[x,y]^s. Thus,(a)if φ(u, v)= 0 for any u,v∈L, then L■Z(R);(b) if φ(u,v)∈ Z(R) for any u,v∈L, then either L■Z(R) or R satisfies S4, the standard identity of degree 4. We also extend the results to semiprime rings.展开更多
Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the sta...Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.展开更多
Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ide...Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ideal of R and n≥1 a fixed positive integer.Under appropriate conditions we prove that if(F(x)x;—xG(x))^n=0 for all x∈L,then one of the following holds:(a)there exists c∈Qr such that F(x)=xc and G(e)=cx;(b)R satisfies S4 and there exist a,b,c∈Qr such that F(x)=ax+xc,G(x)=cx+xb and(a—b)^2=0.展开更多
文摘Let R be a prime ring of characteristic different from 2,Q be its maximal right ring of quotients,and C be its extended centroid.Suppose that f(x1,...,xn)is a non-central multilinear polynomial over C,0≠p∈R,and F,G are two b-generalized derivations of R.In this paper we describe all possible forms of F and G in the case pG(F(f(r))f(r))=0 for all r=(r1,...,rn)in R^n.
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.
文摘Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t≥> 1 fixed integers with t ≤ m + n + s. Suppose that a is a non-trivial automorphism of R and let φ(x,y)=[x,y]^t -[x,y]^m[α([x,y]),[x,y]^n[x,y]^s. Thus,(a)if φ(u, v)= 0 for any u,v∈L, then L■Z(R);(b) if φ(u,v)∈ Z(R) for any u,v∈L, then either L■Z(R) or R satisfies S4, the standard identity of degree 4. We also extend the results to semiprime rings.
文摘Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.
文摘Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ideal of R and n≥1 a fixed positive integer.Under appropriate conditions we prove that if(F(x)x;—xG(x))^n=0 for all x∈L,then one of the following holds:(a)there exists c∈Qr such that F(x)=xc and G(e)=cx;(b)R satisfies S4 and there exist a,b,c∈Qr such that F(x)=ax+xc,G(x)=cx+xb and(a—b)^2=0.
