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
The aim of this paper is to define the notions of generalized （m, n）-derivations and generalized （m, n）：Jordan derivations and to prove two theorems involving these map- pings.
This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identi...This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.展开更多
Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized p...Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.展开更多
In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Fi...In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.展开更多
Let n be a fixed integer, let R be an （n ＋ 1）!-torsion free semiprime ring with the identity element and let F ： R → R be an additive mapping satisfying the relation F（xn＋2） ∑＋n＋i=1xi-1F（x）2xn＋1-i -∑xnF...Let n be a fixed integer, let R be an （n ＋ 1）!-torsion free semiprime ring with the identity element and let F ： R → R be an additive mapping satisfying the relation F（xn＋2） ∑＋n＋i=1xi-1F（x）2xn＋1-i -∑xnF（x）xn＋1-i for all x 6 R. In this case, we prove that F is of the form 2F（x） = D（x） ＋ ax ＋ xa for all x ∈ R, where D ： R → R is a derivation and a 6 R is some fixed element.展开更多
We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for e...We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for each , there exist and e an idempotent of C such that xz=eq.展开更多
In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized （θ,φ）-derivations on a nonzero ideal of a semiprime ring are presented.These results ar...In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized （θ,φ）-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.展开更多
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.
文摘The aim of this paper is to define the notions of generalized （m, n）-derivations and generalized （m, n）：Jordan derivations and to prove two theorems involving these map- pings.
基金Partially supported by China Postdoctoral Science Foundation
文摘This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.
文摘Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.
文摘In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.
文摘Let n be a fixed integer, let R be an （n ＋ 1）!-torsion free semiprime ring with the identity element and let F ： R → R be an additive mapping satisfying the relation F（xn＋2） ∑＋n＋i=1xi-1F（x）2xn＋1-i -∑xnF（x）xn＋1-i for all x 6 R. In this case, we prove that F is of the form 2F（x） = D（x） ＋ ax ＋ xa for all x ∈ R, where D ： R → R is a derivation and a 6 R is some fixed element.
文摘We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for each , there exist and e an idempotent of C such that xz=eq.
文摘In this paper,necessary and sufficient conditions concerning the orthogonality and the composition of a couple of generalized （θ,φ）-derivations on a nonzero ideal of a semiprime ring are presented.These results are generalizations of several results of Breˇsar and Vukman,which are related to a theorem of Posner on the product of two derivations on a prime ring.
