A map $ϕ$ on an associative ring is called a multiplicative Lie derivation if $ϕ([x,y])=[ϕ(x),y]+[x,ϕ(y)]$ holds for any elements $x,y$, where $[x,y]=xy-yx$ is the Lie product. In the paper, we discuss the multiplicative Lie derivations on the triangular 3-matrix rings $\mathcal T={\mathcal T}_3(\mathcal R_i; \mathcal M_{ij})$. Under the standard assumption $Q_i\mathcal Z(\mathcal T)Q_i=\mathcal Z(Q_i\mathcal T Q_i)$, $i=1,2,3$, we show that every multiplicative Lie derivation $φ:\mathcal T\to\mathcal T$ has the standard form $φ=δ+γ$ with $δ$ a derivation and $γ$ a center valued map vanishing each commutator.
In this paper we extend to triangular n-matrix rings and Lie n-multiplicative map a result about Lie multiplicative maps on triangular algebras due to Xiaofei Qi and Jinchuan Hou.
We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive.
Motivated by the works of Wang [Y. Wang, \textit{Lie (Jordan) derivations of arbitrary triangular algebras,} Aequationes Mathematicae, \textbf{93} (2019), 1221-1229] and Moafian et al. [F. Moafian and H. R. Ebrahimi Vishki, \textit{Lie higher derivations on triangular algebras revisited,} Filomat, \textbf{30}(12) (2016), 3187-3194.], we shall study Lie higher derivations of arbitrary triangular algebras. In fact, it is shown that every Lie higher derivation on an arbitrary triangular algebra is proper, using the notion of maximal left (right) ring of quotients.
In this paper we prove that any nonlinear Jordan derivation on triangular algebras is an additive derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert spaces is inner.
In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the triv…
As originally defined by Ashraf and Mozumder, multiplicative (generalized)-skew derivations must satisfy two identities. In this short note we show that, as a consequence of the simultaneous satisfaction of both identities, a multiplicative (generalized)-skew derivation of a prime ring is either a multiplicative (generalized) derivation (i.e., not skew), or a generalized skew derivation (i.e., additive). Therefore only one of the identities should be taken in the definition of multiplicative (generalized)-skew derivations in order to get a new class of derivations in prime rings.
We provide that any Jordan derivation from the block upper triangular matrix algebra $\T = \T(n_{1},n_{2}, \cdots, n_{k})\subseteq M_{n}(\mathbb{\C})$ into a $2$-torsion free unital $\T$-bimodule is the sum of a derivation and an antiderivation.
We explore Jordan derivations of triangular matrices with entries from an additively idempotent semiring. The main result states that for any matrix A over additively idempotent semiring, if we put all the elements of the family of dense submatrices of A to be zeroes, we find a derivative of A. The set of derivations of this type is established.
Let N be a left near ring. A map d on N is called a nonzero multiplicative derivation if d(xy)=xd(y)+d(x)y holds for all x,y elements of N.In the present paper, we shall extend some well known results concerning commutativity of prime rings for a nonzero multiplicative derivations of a left prime near-ring N.
We generalize the results of Leger and Luks, Zhang R. and Zhang Y.; Chen, Ma, Ni, Niu, Zhou and Fan; Kaygorodov and Popov about generalized derivations of color $n$-ary algebras to the case of $n$-ary Hom-$Ω$ color algebras. Particularly, we prove some properties of generalized derivations of multiplicative $n$-ary Hom-$Ω$ color algebras. Moreover, we prove that the quasiderivation algebra of any multiplicative $n$-ary Hom-$Ω$ color algebra can be embedded into the derivation algebra of a larger multiplicative n-ary Hom-$Ω$ color algebra.
Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.
Guided by the research line introduced by Martindale III in [1] on the study of the additivity of maps, this article aims establish condi- tions on triangular matrix rings in order that an map ' satisfying '(ab + ba) = '(a)b + a'(b) + '(b)a + b'(a) for all a; b in a triangular matrix ring becomes additive.
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
We explicitly describe the Lie algebras $M_L$ of ladder matrices in $M_n$ associate with dominant upper triangular ladders $L$, and completely characterize the derivations of these $M_L$ over a field $F$ with $char(F) \neq 2$. We also completely characterize the derivations of Lie algebras $[M_L, M_L]$ where $L$ are strongly dominant upper triangular ladders and $char(F) \neq 2, 3$.
In this paper, we introduce the relevant concepts of $n$-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of $n$-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations for $n$-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation of $n$-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.
We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.
Let ${\mathcal N}$ be the Lie algebra of all $n\times n$ strictly block upper triangular matrices over a field ${\mathbb F}$ relative to a given partition. In this paper, we give an explicit description of all derivations of ${\mathcal N}$.
Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $Δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More precisely, it is proved that $Δ$ is of the form $Δ(t)=μt+δ(t),$ where $μ\in\mathcal{Z}(\mathfrak{A})$ and $δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is a Jordan $n$-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, prime rings, nest algebras, standard operator a…
It is well known that n-Hom Lie superalgebras are certain generalizations of n-Lie algebras. This paper is devoted to investigate the generalized derivations of multiplicative n-Hom Lie superalgebras. We generalize the main results of Leger and Luks to the case of multiplicative n-Hom Lie superalgebras. Firstly, we review some concepts associated with a multiplicative n-Hom Lie superalgebra $N$. Furthermore, we give the definitions of the generalized derivations, quasiderivations, center derivations, centroids and quasicentroids. Obviously, we have the following tower $ZDer(N)\subseteq Der(N)\…
The aim of this paper is introduce and give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n- BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras.
Let $\T$ be a $2$-torsion free triangular ring and let $φ:\T\rightarrow \T$ be an additive map. We prove that if $\A φ(\B)+φ(\B)\A=0$ whenever $\A,\B\in \T$ are such that $\A\B=\B\A=0$, then $φ$ is a centralizer. It is also shown that if $τ:\T\rightarrow \T$ is an additive map satisfying $\label{t2} X,Y\in \T, \quad XY=YX=0\Rightarrow X τ(Y)+δ(X)Y+Yδ(X)+τ(Y)X=0$, where $δ:\T\rightarrow \T $ is an additive map satisfies $X,Y\in \T, \quad XY=YX=0\Rightarrow X δ(Y)+δ(X)Y+Yδ(X)+δ(Y)X=0$, then $τ(\A)=d(\A)+\A τ(\textbf{1})$, where $d:\T\rightarrow \T$ is a derivation and $τ(\textbf{1})$ lies in the…
In this article we construct examples of derivations in matrix semirings. We study hereditary and inner derivations, derivatives of diagonal, triangular, Toeplitz, circulant matrices and of matrices of other forms and prove theorems for derivatives of matrices of these forms.
Let $\R$ be an alternative ring containing a nontrivial idempotent and $\D$ be a multiplicative Lie-type derivation from $\R$ into itself. Under certain assumptions on $\R$, we prove that $\D$ is almost additive. Let $p_n(x_1, x_2, \cdots, x_n)$ be the $(n-1)$-th commutator defined by $n$ indeterminates $x_1, \cdots, x_n$. If $\R$ is a unital alternative ring with a nontrivial idempotent and is $\{2,3,n-1,n-3\}$-torsion free, it is shown under certain condition of $\R$ and $\D$, that $\D=δ+τ$, where $δ$ is a derivation and $τ\colon\R\longrightarrow{\mathcal Z}(\R)$ such that $τ(p_n(a_1,\ldots,…
The main purpose of this paper is to show that every Jordan centralizer and every Jordan two-sided centralizer is a centralizer on triangular rings without assuming unity. As an application, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. Some other related results are also discussed.
Let $A$ be an algebra and $σ$ an automorphism of $A$. A linear map $d$ of $A$ is called a $σ$-derivation of $A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in A$. A linear map $D$ is said to be a generalized $σ$-derivation of $A$ if there exists a $σ$-derivation $d$ of $A$ such that $D(xy) = D(x)y + σ(x)d(y)$, for all $x, y \in A$. An additive map $Θ$ of $A$ is $σ$-centralizing if $Θ(x)x - σ(x)Θ(x) \in Z(A)$, for all $x \in A$. In this paper, precise descriptions of generalized $σ$-derivations and $σ$-centralizing maps of triangular algebras are given. Analogues of the so-called commutative …
Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $δ:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ satisfies that $δ(xy)=δ(x)y+xδ(y)$ for any two rank-$s$ matrices $x,y\in M_n(\mathbb{K})$, then there exists a derivation $D$ of $M_n(\mathbb{K})$ such that $δ(x)=D(x)$ holds for each rank-$k$ matrix $x\in M_n(\mathbb{K})$ with $0\leq k\leq s$.
In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be decomposed into the sum of a derivation and a center valued map. We extend some known results on the properness of Lie derivations of triangular algebras. Some illuminating examples are also included.
Let J and J' be Jordan rings. We prove under some conditions that if J contains a nontrivial idempotent, then n-multiplicative maps and n-multiplicative derivations from J to J' are additive maps.
Let C be a commutative ring with unity. In this article, we show that every Jordan derivation over an upper triangular matrix algebra T_n(C) is an inner derivation. Further, we extend the result for Jordan derivation on full matrix algebra M_n(C).
It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebra is the maximal Lie algebra which lies in the set of locally nilpotent derivations of the ring. Also it was conjectured that each other maximal Lie algebra which lies in the set of locally nilpotent derivations of the polynomial ring is conjugated to the triangular Lie algebra. In the present work we prove the first part of this conjecture and provide the counterexample to the second part. Also we show that the second part of the conjecture holds for the maximal Lie algebras among locally nilpoten…
Let $\A$ be an algebra and $σ$ an automorphism of $\A$. A linear map $d$ of $\A$ is called a $σ$-derivation of $\A$ if $d(xy) = d(x)y + σ(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$ is said to be a $σ$-biderivation of $\A$ if it is a $σ$-derivation in each component. An additive map $Θ$ of $\A$ is $σ$-commuting if it satisfies $Θ(x)x - σ(x)Θ(x) = 0$, for all $x \in \A$. In this paper, we introduce the notions of inner and extremal $σ$-biderivations and of proper $σ$-commuting maps. One of our main results states that, under certain assumptions, every $σ$-biderivatio…
We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several characterizations, such as the existence of gradings by sixth roots of unity, or the existence of a nonsingular derivation whose inverse is again a derivation. We also obtain results on the existence of periodic prederivations. In this context we study a generalization of Engel-4-Lie algebras.
The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative super-algebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.
Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not necessarily with the identity elements). A bijective map $φ$ of $\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative isomorphism} if {$φ(x_{1} \cdots x_{m}) = φ(x_{1}) \cdots φ(x_{m})$} for all $x_{1}, \cdots ,x_{m}\in \mathfrak{R}.$ In this article, we establish a condition on generalized $n$-matrix rings, that assures that multiplicative maps are additive on generalized $n$-matrix rings under certain restrictions. And then, we apply our result for study of $m$-multiplicative isomorphism and $m$-multipl…
In this paper, we introduce the notion of a BiHom-Lie conformal algebra and develop its cohomology theory. Also, we discuss some applications to the study of deformations of regular BiHom-Lie conformal algebras. Finally, we introduce derivations of multiplicative BiHom-Lie conformal algebras and study their properties.
In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether the algebra admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nilpotent dimensionally nilpotent Lie triple algebras. We show that when $n=2p+1$ the adapted basis coincides with the canonical basis of the gametic algebra $G(2p+2,2)$ or this one obviously associated to a pseudo-idempotent and if $n=2p$ then the algebra is either one of the precedent case or a conservative Bernstein algebra.
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.
In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.
Let $(\mathfrak{g},ω)$ be a finite-dimensional non-Lie complex $ω$-Lie algebra. We study the derivation algebra $Der(\mathfrak{g})$ and the automorphism group $Aut(\mathfrak{g})$ of $(\mathfrak{g},ω)$. We introduce the notions of $ω$-derivations and $ω$-automorphisms of $(\mathfrak{g},ω)$ which naturally preserve the bilinear form $ω$. We show that the set $Der_ω(\mathfrak{g})$ of all $ω$-derivations is a Lie subalgebra of $Der(\mathfrak{g})$ and the set $Aut_ω(\mathfrak{g})$ of all $ω$-automorphisms is a subgroup of $Aut(\mathfrak{g})$. For any 3-dimensional and 4-dimensional nontrivial $ω$-L…
We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n \times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. In the $2 \times 2$ case this also yields a representation as a subsemigroup of a semidirect product of commutative monoids. In particular, from the case where $n=2$ and the semiring is the tropical semifield, we obtain a representation of the free objects in the monoid and semigroup varieties gener…
We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for $2$-step nilpotent Lie algebras determined by graphs, free $2$ and $3$-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand we also exhibit famili…
The aim of this paper is to provide cohomologies of $n$-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to $n$-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of $n$-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.
Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of $LND(A)$ is of great interest, and the question about properties of Lie algebras contained in $LND(A)$ is still open. An answer to it in the finite dimensional case is given. It is proved that any finite dimensional (over $K$) subalgebra of $Der_{K}(A)$ consisting of locally nilpotent derivations is nilpotent. In the case $A=K[x, y],$ it is also proved that any …
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ is a functor from the category of $k$-algebras with pre-morphisms to the category of anticommutative $k$-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This a…
In commutative algebra, if $δ$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $δ$, then $Δ=wδ$ is also a locally nilpotent derivation with the same kernel as $δ$. In this paper we prove that the locally nilpotent derivation $Δ$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $Δ$ is a free associative algebra and give an explicit set of its free generators.
Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-$p$ groups and Lie algebras. A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of positive characteristic p was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k-1 with $n^4>(2^k-n)^{3}$ belongs to N_2. Our methods consist of elementary argumen…
