Xiao Wei Xu

• Jilin University

Let (Formula presented.) be integers, and let (Formula presented.) be the ring of all (Formula presented.) matrices over a field (Formula presented.) with centre (Formula presented.). Assume that (Formula presented.) is an m-additive map such that (Formula presented.) ((Formula presented.)), where (Formula presented.) denotes the trace map of G and...

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\l...

Let R be a prime ring with extended centroid C. In this paper, we discuss the case when the composition of a generalized derivation δ and a polynomial map f(Y) ∈ C[Y] of R is commutative on a non-zero right ideal ρ and a non-commutative Lie ideal L of R respectively, i.e., when the identity δ ○ f(x) = f ○ δ(x) holds on ρ or L. As applications of ou...

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal and a noncommutative Lie...

In this paper, firstly as a short note, we prove that a left derivation of a semiprime $\Gamma$-ring $M$ must map $M$ into its center, which improves a result by Paul and Halder and some results by Asci and Ceran. Also we prove that a semiprime $\Gamma$-ring with a strong commutativity preserving derivation on itself must be commutative and that a...

In 1969, Martindale proved a famous theorem well known as the Martindale lemma. In this paper, we give a generalization of the Martindale lemma by considering the nonlinear case. © 2012 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.

Let R be a prime ring with extended centroid C and maximal right ring of quotients U. Let δ be a generalized derivation of R, I a nonzero ideal of R and a,b,q∈U. In this paper, we discuss the identity (aδ(qx)-bx) n =0 (∈C) for all x∈I through which we give a generalization of the results obtained by I. N. Herstein, [J. Algebra 60, 567-574 (1979; Zb...

In this paper we describe generalized left (Θ, Φ)-derivations in prime rings, and prove that an additive mapping in a ring R acting as a homomorphism or anti-homomorphism on an additive subgroup S of R must be either a mapping acting as a homomorphism on S or a mapping acting as an anti-homomorphism on S, through which some related results are impr...

In this paper we describe generalized left (theta, phi)-derivations in prime rings, and prove that an additive mapping in a ring R acting as a homomorphism or anti-homomorphism on an additive subgroup S of R must be either a mapping acting as a homomorphism on S or a mapping acting as an anti-homomorphism on S, through which some related results ar...

Left derivations and Jordan left derivations in full and upper triangular matrix rings over unital associative rings are characterized.

Let R be a prime ring with extended centroid C, a nonzero ideal I and two derivations d1, d2. Suppose that d1(x)ⁿ = d2(x)ⁿ for all x ∈ I. Then there exists λ ∈ C such that d2(x) = λd1(x) for all x ∈ R.

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativity-preserving (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 als...

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.