In this article, we introduce the notion of generalized derivations on Hilbert C*-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen...In this article, we introduce the notion of generalized derivations on Hilbert C*-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen type functional equationrf(x+y/r)+sg(x-y/s)=2h(x)for r, s ∈ R / {0} on Hilbert C*-modules, where f, g, and h are mappings from a Hilbert C*-module M to M.展开更多
Let P be a parabolic subalgebra of a general linear Lie algebra gl(n,F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) ≠ 2. In this article, we prove that generalized derivati...Let P be a parabolic subalgebra of a general linear Lie algebra gl(n,F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.展开更多
This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al.
Let R be a prime ring with center Z and S包含 R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x - xG(x) ∈ Z for all x ∈S. The main purpose of this paper is to describe the structur...Let R be a prime ring with center Z and S包含 R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x - xG(x) ∈ Z for all x ∈S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime ease is also considered.展开更多
Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b i...Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.展开更多
Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibil...Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.展开更多
Let U = Tri(fit, M, B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map φ on U is centralized at zero point (i.e., ,φ(A)B = A,φ(B)...Let U = Tri(fit, M, B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map φ on U is centralized at zero point (i.e., ,φ(A)B = A,φ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let 5 : U →U be an additive map. It is also shown that the following four conditions are equivalent: (1) 5 is specially generalized derivable at zero point, i.e., 5(AB) = δ(A)B + AS(B) - Aδ(I)B whenever AB = 0; (2) 5 is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1(B) = τ2(A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space展开更多
Let R be a semiprime ring, R F be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of R F . Fur...Let R be a semiprime ring, R F be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of R F . Furthermore, if there exists a fixed positive integer n such that μ(x) n = 0 for all x ∈I, then μ = 0.展开更多
Let R be a prime ring of characteristic different from 2 with Utumi quotientring U and extended centroid C,F and G,the two nonzero generalized derivationsof R,I an ideal of R and f(x1,...,xn)a multilinear polynomial o...Let R be a prime ring of characteristic different from 2 with Utumi quotientring U and extended centroid C,F and G,the two nonzero generalized derivationsof R,I an ideal of R and f(x1,...,xn)a multilinear polynomial over C which is notcentral valued on R.If F(G(f(x1,...,Xn))f(X1,...,Xn))=0 for all x1,...,Xn∈1,then one of the followings holds:(1)there exist a,b c Usuch that F(x)=ax and G(x)=bx for all x c R with ab=0;(2)there exista,b,p c U such that F(x)=ax+xb and G(x)=px for all x c R with F(p)=0and f(x1,...,xn)’is central valued on R.We also obtain some related results in caseswhere R is a semiprime ring and Banach algebra.展开更多
Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving (scp) on S, if [f(x), f(y)] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the stru...Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving (scp) on S, if [f(x), f(y)] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.展开更多
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie con...The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.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 AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, t...Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.展开更多
The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutativ...The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char(R) ≠ 2. If R admits a generalized derivation F : R → R associated with a derivation d : R → R such that [F(x),F(x*)] - [x,x*] = 0 for all x ∈ R, then F(x)= x for all x ∈ R or F(x) = -x for all x ∈ R. Moreover, a related result is also obtained.展开更多
The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R → R satisfying δ (xy) = δ(x)y+xd(y) for all x, y ∈ R, where d is a derivation on R. Such a fu...The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R → R satisfying δ (xy) = δ(x)y+xd(y) for all x, y ∈ R, where d is a derivation on R. Such a function δ is called a generalized derivation. Suppose that U is a Lie ideal of R such that u^2 ∈ U for all u ∈ U. In this paper, we prove that U lahtain in Z(R) when one of the following holds: (1) δ([u, v]) = u o v =(2) δ([u,v])=[u o v] = 0 (3) δ(u o v) = [u, v] (4) δ(u o v)+δ[u, v] = 0 for all u, v ∈ U.展开更多
Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. U...Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. Using it, we show that the separability of an algebra extension is characterized by generalized derivations.展开更多
We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provid...We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provide all generalized derivations for complex low-dimensional dendriform algebras.展开更多
Let R be a prime ring and n 〉 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)n = g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2...Let R be a prime ring and n 〉 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)n = g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2 matrix ring over a field. Moreover, we generalize the result to the case g(f(xi))n = g(f(xi)) for all xl,x2,...,xt ∈ R, where f(Xi) is a multilinear polynomial.展开更多
In this paper, it is proved that under certain conditions, each Jordan left derivation on a generalized matrix algebra is zero and each generalized Jordan left derivation is a generalized left derivation.
With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Sehrodinger equation (GDNLSE) hav...With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Sehrodinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.展开更多
文摘In this article, we introduce the notion of generalized derivations on Hilbert C*-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen type functional equationrf(x+y/r)+sg(x-y/s)=2h(x)for r, s ∈ R / {0} on Hilbert C*-modules, where f, g, and h are mappings from a Hilbert C*-module M to M.
基金supported by the National Natural Science Foundation of China(11101084,11071040)the Fujian Province Nature Science Foundation of China(2013J01005)
文摘Let P be a parabolic subalgebra of a general linear Lie algebra gl(n,F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.
文摘This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al.
文摘Let R be a prime ring with center Z and S包含 R. Two mappings D and G of R into itself are called cocentralizing on S if D(x)x - xG(x) ∈ Z for all x ∈S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime ease is also considered.
文摘Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.
基金The first author supported in part by NNSF(10726051)of ChinaGrant in-aid for Scientific Research from Department of Mathematics,Jilin UniversityThe second author supported by Grant in-aid for Scientific Research from Department of Mathematics,Jilin University.
文摘Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.
基金supported by National Natural Science Foundation of China (Grant No. 11101250)supported by National Natural Science Foundation of China (Grant No. 11171249)Youth Foundation of Shanxi Province (Grant No. 2012021004)
文摘Let U = Tri(fit, M, B) be a triangular ring, where A and B are unital rings, and M is a faithful (A, B)-bimodule. It is shown that an additive map φ on U is centralized at zero point (i.e., ,φ(A)B = A,φ(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let 5 : U →U be an additive map. It is also shown that the following four conditions are equivalent: (1) 5 is specially generalized derivable at zero point, i.e., 5(AB) = δ(A)B + AS(B) - Aδ(I)B whenever AB = 0; (2) 5 is generalized derivable at zero point, i.e., there exist additive maps τ1 and τ2 on U derivable at zero point such that δ(AB) = δ(A)B + Aτ1(B) = τ2(A)B + Aδ(B) whenever AB = 0; (3) δ is a special generalized derivation; (4) δ is a generalized derivation. These results are then applied to nest algebras of Banach space
基金supported by the mathematical Tianyuan research foundationthe post-doctorate research foundation
文摘Let R be a semiprime ring, R F be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of R F . Furthermore, if there exists a fixed positive integer n such that μ(x) n = 0 for all x ∈I, then μ = 0.
基金The authors would like to thank the referee for providing shortened proof of Lemma2.1 in the paper.This work was done when the first author visited Ege University,TURKEY,from the9th June 2014 to the 15th June 2014 under the INSA-TUBA Exchange of Scientists Programme.The firstauthor is grateful to INSA,India and TUBA,Turkey for the financial support provided for this visit.
文摘Let R be a prime ring of characteristic different from 2 with Utumi quotientring U and extended centroid C,F and G,the two nonzero generalized derivationsof R,I an ideal of R and f(x1,...,xn)a multilinear polynomial over C which is notcentral valued on R.If F(G(f(x1,...,Xn))f(X1,...,Xn))=0 for all x1,...,Xn∈1,then one of the followings holds:(1)there exist a,b c Usuch that F(x)=ax and G(x)=bx for all x c R with ab=0;(2)there exista,b,p c U such that F(x)=ax+xb and G(x)=px for all x c R with F(p)=0and f(x1,...,xn)’is central valued on R.We also obtain some related results in caseswhere R is a semiprime ring and Banach algebra.
基金China NNSF (10726051)Grant in-aid for Scientific Research from Department of Mathematics,Jilin University
文摘Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving (scp) on S, if [f(x), f(y)] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171055, 11471090 and 11301109)Natural Science Foundation of Jilin Province (Grant No. 20170101048JC)
文摘The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.
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.
基金Supported by the National Natural Science Foundation of China(Grant No.11571247)supported by the union program of department of science technology in Guizhou province,Anshun government and Anshun university(Grant No.201304)
文摘Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.
文摘The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char(R) ≠ 2. If R admits a generalized derivation F : R → R associated with a derivation d : R → R such that [F(x),F(x*)] - [x,x*] = 0 for all x ∈ R, then F(x)= x for all x ∈ R or F(x) = -x for all x ∈ R. Moreover, a related result is also obtained.
文摘The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R → R satisfying δ (xy) = δ(x)y+xd(y) for all x, y ∈ R, where d is a derivation on R. Such a function δ is called a generalized derivation. Suppose that U is a Lie ideal of R such that u^2 ∈ U for all u ∈ U. In this paper, we prove that U lahtain in Z(R) when one of the following holds: (1) δ([u, v]) = u o v =(2) δ([u,v])=[u o v] = 0 (3) δ(u o v) = [u, v] (4) δ(u o v)+δ[u, v] = 0 for all u, v ∈ U.
文摘Recently, we introduced the notion of a generalized derivation from a bimodule to a bimodule. In this paper, we give a more general notion based on commutators which covers generalized derivations as a special case. Using it, we show that the separability of an algebra extension is characterized by generalized derivations.
文摘We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provide all generalized derivations for complex low-dimensional dendriform algebras.
文摘Let R be a prime ring and n 〉 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)n = g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2 matrix ring over a field. Moreover, we generalize the result to the case g(f(xi))n = g(f(xi)) for all xl,x2,...,xt ∈ R, where f(Xi) is a multilinear polynomial.
基金Fundamental Research Funds (N110423007) for the Central Universities
文摘In this paper, it is proved that under certain conditions, each Jordan left derivation on a generalized matrix algebra is zero and each generalized Jordan left derivation is a generalized left derivation.
基金the Natural Science Foundation of Education Department of Henan Province of China under Grant No.2007110010the Science Foundation of Henan University of Science and Technology under Grant Nos.2006ZY-001 and 2006ZY-011
文摘With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Sehrodinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.