A note on commuting additive maps on rank k symmetric matrices

Let $n\geq 2$ and $1<k\leq n$ be integers. Let $S_n(\mathbb{F})$ be the linear space of $n\times n$ symmetric matrices over a field $\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\psi:S_n(\mathbb{F})\rightarrow S_n(\mathbb{F})$ satisfies $\psi(A)A=A\psi(A)$ for all rank $k$ matrices $A\in S_n(\mathbb{F})$ if and only if there exists a scalar $\lambda\in \mathbb{F}$ and an additive map $\mu:S_n(\mathbb{F})\rightarrow \mathbb{F}$ such that\[\psi(A)=\lambda A+\mu(A)I_n,\]for all $A\in S_n(\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\mathbb{F}$ of characteristic not two are included.

Stability results and separations theorems

The presented work summarizes the relationships between stability results and separation theorems. We prove the equivalence between different types of theorems on separation by an additive map and different types of stability results concerning the stability of the Cauchy functional equation.

Characterizations of additive -Lie derivations on unital algebras

UDC 512.5 Let be a commutative ring with unity and be a unital algebra over (or field ).An -linear map is called a Lie derivation on if holds for all For scalar an additive map is called an additive -Lie derivation on if where holds for all In the present paper, under certain assumptions on it is shown that every Lie derivation (resp., additive -Lie derivation) on is of standard form, i.e., where is an additive derivation on and is a mapping vanishing at with in Moreover, we also characterize the additive -Lie derivation for by its action at zero product in a unital algebra over

Additive Maps of Rank k Bivectors

Let ${\cal U}$ and ${\cal V}$ be linear spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively, such that Dim$\,{\cal U}=n\geqslant 2$ and $\left|\mathbb{F}\right|\geqslant 3$. Let $\bigwedge^2{\cal U}$ be the second exterior power of ${\cal U}$. Fixing an even integer $k$ satisfying $\frac{n-1}{2}\leqslant k\leqslant n$, it is shown that a map $\psi:\bigwedge^2{\cal U}\rightarrow\bigwedge^2{\cal V}$ satisfies $\psi(u+v)=\psi(u)+\psi(v)$ for all rank $k$ bivectors $u,v\in\bigwedge^2{\cal U}$ if and only if $\psi$ is an additive map. Examples showing the indispensability of the assumption on $k$ are given.

Additive n-commuting maps on semiprime rings

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

m-commuting Additive Maps on Upper Triangular Matrices Rings

Let $T_{n}(R)$ be the upper triangular matrix ring over a unital commutative ring whose characteristic is not a divisor of $m$. Suppose that $f:T_{n}(R)\rightarrow T_{n}(R)$ is an additive map such that $X^{m}f(X)=f(X)X^{m}$ for all $x \in T_{n}(R),$ where $m\geq 1$ is an integer. We consider the problem of describing the form of the map $X \rightarrow f(X)$.

Multiplicative (Generalized) Reverse Derivations on Semiprime Ring

Let R be a semiprime ring. A mapping F : R → R (not necessarily additive) is called a multiplicative (generalized) reverse derivation if there exists a map d : R → R (not necessarily a derivation nor an additive map) such that F(xy) = F(y)x + yd(x) for all x, y є R. In this paper we investigate some identities involving multiplicative (generalized) reverse derivation and prove some theorems in which we characterize these mappings.

Additive Maps on Units of Rings

Let R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

The commutativity of prime Γ-rings with generalized skew derivations

Let M be a prime Γ-ring with center {Z(M)}, and let θ be an automorphism of M. An additive map {d:M\to M} is called a skew derivation if {d(x\alpha y)=d(x)\alpha y+\theta(x)\alpha d(y)} for all {x,y\in M}, {\alpha\in\Gamma}. An additive map {F:M\to M} is called a generalized skew derivation if there exists a skew derivation {d:M\to M} such that {F(x\alpha y)=F(x)\alpha y+\theta(x)\alpha d(y)} holds for all {x,y\in M}, {\alpha\in\Gamma}. In the present paper, our main objective is to prove some commutativity results for prime Γ-rings M admitting a generalized skew derivation F satisfying anyone of the properties:(i){F(x\alpha y)\pm x\alpha y\in Z(M)},(ii){F(x\alpha y)\pm y\alpha x\in Z(M)},(iii){F(x)\alpha F(y)\pm x\alpha y\in Z(M)},(iv){F([x,y]_{\alpha})\pm[x,y]_{\alpha}=0},(v){F(\langle x,y\rangle_{\alpha})\pm\langle x,y\rangle_{\alpha}=0}for all {x,y\in I} and {\alpha\in\Gamma}. In fact, we obtain rather more general results which unify, extend and complement several well-known results proved in [3, 4, 5, 6, 32].

Additive maps having a generalized derivation expansion

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.