In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$, then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$-subspace lattice algebras, is a derivation.

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

The purpose of this note is to prove the following. Suppose $\mathfrak{R}$ is a semiprime unity ring having an idempotent element $e$ ($e\neq 0,~e\neq 1$) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\mathfrak{R}$ is a generalized derivation.

We determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

Let $R$ be a prime ring of characteristic diòerent from $2$, let ${{Q}_{r}}$ be its right Martindale quotient ring, and let $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R,\,L$ a non-central Lie ideal of $R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$ and $n,\,s\,\ge \,1$ fixed integers. If

$$a{{\left( {{u}^{m}}F\left( u \right){{u}^{n}} \right)}^{s}}\,=\,0$$

for all $u\,\in \,L$, then either $R\,\subseteq \,{{M}_{2}}\left( C \right)$, the ring of $2\,\times \,2$ matrices over $C$, or $m\,=\,0$ and there exists $b\,\in \,{{Q}_{r}}$ such that $F\left( x \right)\,=\,bx$, for any $x\,\in \,R$, with $ab\,=\,0$.

Let $R$ be a ring and let $g$ be an endomorphism of $R$. The additive mapping $d:\,R\,\to \,R$ is called a Jordan semiderivation of $R$, associated with $g$, if

$$d\left( {{x}^{2}} \right)=d\left( x \right)x+g\left( x \right)d\left( x \right)=d\left( x \right)g\left( x \right)+xd\left( x \right)\,\text{and}\,d\left( g\left( x \right) \right)=g\left( d\left( x \right) \right)$$

for all $x\,\in \,R$. The additive mapping $F:\,R\,\to \,R$ is called a generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if

$$F\left( {{x}^{2}} \right)=F\left( x \right)x+g\left( x \right)d\left( x \right)=F\left( x \right)g\left( x \right)+xd\left( x \right)\,\,and\,F\left( g\left( x \right) \right)=g\left( F\left( x \right) \right)$$

for all $x\,\in \,R$. In this paper we prove that if $R$ is a prime ring of characteristic different from 2, $g$ an endomorphism of $R,\,d$ a Jordan semiderivation associated with $g,\,F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R$. Moreover, if $R$ is commutative, then $F\,=\,d$.

Let A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a semiprime ring with extended centroid $C$ and with maximal right ring of quotients $Q_{mr}(R)$. Let $d{:}\ R\to Q_{mr}(R)$ be an additive map and $b\in Q_{mr}(R)$. An additive map $\delta {:}\ R\to Q_{mr}(R)$ is called a (left) $b$-generalized derivation with associated map $d$ if $\delta (xy)=\delta (x)y+bxd(y)$ for all $x, y\in R$. This gives a unified viewpoint of derivations, generalized derivations and generalized $\sigma $-derivations with an X-inner automorphism $\sigma $. We give a complete characterization of $b$-generalized derivations of $R$ having nilpotent values of bounded index. This extends several known results in the literature.

In this paper, we characterize Jordan $*$-derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $. Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$-derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, char $R\,\ne \,2$, and let $\delta $ be a nonzero Jordan $*$-derivation of $R$. If $\delta $ is an elementary operator of $R$, then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

For a derivation δ of a commutative Noetherian ${\mathbb C}$-algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;δ] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=δ(a) for all a ∈ A.

It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Generalizing a result of Yoshinaga in dimension three, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds:

1. (1) aI=ag(I)=(0);

2. (2) g(x)=qx, for some q∈U and aqI=0;

3. (3) [f(x1,…,xn),xn+1]xn+2 is an identity for I;

4. (4) g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.

The aim of this note is to characterize 2-local automorphisms and derivations on matrix rings over finite-dimensional division rings.

Let $K$ be a field of characteristic zero and $A_{2}:=A_{2}(K)$ the 2$nd$–Weyl algebra over $K$. We establish a close connection between the maximal left ideals of $A_{2}$ and the simple derivations of $K[X_{1},X_{2}]$.

MAIN THEOREM. Let$d = \partial_1 + \beta\partial_2$be a simple derivation of$K[X_1, X_2]$with$\beta\in K[X_1, X_2]$. Then, there exists$\gamma\in K[X_1, X_2]$such that$d + \gamma$generates a maximal left ideal of$A_2$. More precisely, the following is true:

1. $\deg_{X_2} (\beta) \geq 2$ or $\deg_{X_2} (\beta) = 1$ and $\deg_{X_1} (\partial_2(\beta)) \geq 1$;

2. $d + gX_2$generates a maximal left ideal of$A_2$if$g \in K[X_1]\\{0\}$is such that

1. (a) $g \in -(\partial^2_2(\beta)/2)\mathbb{N}$when$\deg_{X_2} (\beta) \geq 2$,

2. (b) $\deg_{X_1} (g) < \deg_{X_1} (\partial_2(\beta))$when$\deg_{X_2} (\beta) = 1$.

As applications, we obtain large families of concrete examples of cyclic maximal left ideals of $A_2$; such examples have been rather scarce so far.

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R, f (x1,…, xn) a multilinear polynomial over K.If

then f(x1,…,xnis central-valued on R.

Let $R$ be a non-$\text{GPI}$ prime ring with involution and characteristic $\ne 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta \,=\,\rho \,+\,\varepsilon$ where $\varepsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $\left[ K,\,K \right]$, and $\rho$ can be extended to an ordinary derivation of $\left\langle K \right\rangle$ into $RC$, the central closure.

Let R be a prime ring with involution and d, δ be derivations on R. Suppose that xd(x)—δ(x)x is central for all symmetric x or for all skew x. Then d = δ = 0 unless R is a commutative integral domain or an order of a 4-dimensional central simple algebra.

Let RG denote the group ring of a group G over a semiprime ring R. We prove that, if the center of G is of finite index and some natural restrictions hold, then every R-derivation of RG is inner. We also give an example of a group G which is both locally finite and nilpotent and such that, for every field F, there exists an F-derivation of FG which is not inner.