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 $τ(p_n(a_1,\ldots,a_n))=0$ for all $a_1,\ldots,a_n\in\R$.
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.
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 $\mathfrak{A}$ 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…
In this paper, we introduce multiplicative semiderivation and we investigate the commutativity of semiprime rings satisfying certain conditions and identities involving multiplicative semiderivations on a nonzero ideal I of a ring R.
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.
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 give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its centre, thus disproving the conjecture posed in the paper "Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8".
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 $\mathfrak{R}$ 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…
Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.
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 $0\leq k\leq s$.
Let D be a division ring with centre F. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In [1], authors have conjectured that every division ring is generated as a vector space over its centre by all of its multiplicative commutators. In this note it is shown that if D is centrally finite, then the conjecture holds.
Let $D$ be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^*$ of $D$. The new obtained results generalize some previous results on subgroups of $D^*$.
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.
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 …
Let $p$ be a prime such that the multiplicative order $m$ of $2$ modulo $p$ is even. We prove that the integral group ring $\mathbb{Z}[Q_8 \times C_p]$ has the multiplicative Jordan decomposition property when $m$ is congruent to $2$ modulo $4$. There are infinitely many such primes and these primes include the case $p \equiv 3 \pmod{4}$. We also prove that $\mathbb{Z}[Q_8 \times C_5]$ has the multiplicative Jordan decomposition property in a new way.
Automorphism groups of direct products of multiplicative monoids of certain rings
University of Stellenbosch（斯泰伦博斯大学）；NITheCS (National Institute for Theoretical and Computational Sciences)（国家理论与计算科学研究所）；University of Minho（米尼奥大学）；CMAT (Centre of Mathematics)（数学中心）
Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center by locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occu…
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-adjoint matrices over a commutative $*$-ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. We also show that every local spatial derivation on the above Lie algebras is a derivation.
Let $P$ be a preordered set, $R$ a ring and $FI(P,R)$ the finitary incidence ring of $P$ over $R$. We find a criterion for all Jordan derivations of $FI(P,R)$ to be derivations and generalize Theorem 3.3 from arXiv:1411.6123. In particular, we prove that each Jordan derivation of the ring $RFM_I(R)$ of row-finite $I\times I$-matrices over $R$ is a derivation, if $|I|>1$.
Let $K$ be a 2-torsion free ring with identity. We give a description of the Lie derivations of $R=R_{n}(K,J)=NT_{n}(K)+M_{n}(J)$, the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are elements of an ideal $J$ of $K$.
We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations t…
The main purpose of this paper is to study and investigate concerning a (α,α)-symmetric derivations D on semiprime rings and prime rings R, we give some results when R admits a (α,α)-symmetric derivations D to satisfy some conditions on R.Where α: R to R is an automorphism mapping.
We study $(σ,τ)$-derivations of a group ring $RG$ of a finite group $G$ over an integral domain $R$ with $1$. As an application we extend a well known result on derivation of an integral group ring $\Bbb{Z}G$ to $(σ,τ)$-derivation on it for a finite group $G$ with some conditions on $σ$ and $τ$. In the process of the extension, a generalization of an application of Skolem-Noether Theorem to derivation on a finite dimensional central simple algebra has also been given for the $(σ,τ)$-derivation case.
The purpose of this note is to prove the following. Suppose $\R$ is a semiprime unity ring having an idempotent element e $\left(e \neq 0, e \neq 1\right)$ which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\R$ is a generalized derivation.
Let $K$ be a 2-torsion free ring with identity and $R_{n}(K,J)$ be the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are elements of an ideal $J$ of $K.$ We describe all Jordan derivations of the matrix ring $R_{n}(K,J)$ in this paper. The main result states that every Jordan derivation $Δ$ of $R_{n}(K,J)$ is of the form $Δ=D+Ω$ where $D$ is a derivation of $R_{n}(K,J)$ and $Ω$ is an extremal Jordan derivation of $R_{n}(K,J).$
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.
For a ring $R$ with an automorphism $σ$, an $n$-additive mapping $Δ:R\times R\times... \times R \rightarrow R$ is called a skew $n$-derivation with respect to $σ$ if it is always a $σ$-derivation of $R$ for each argument. Namely, it is always a $σ$-derivation of $R$ for the argument being left once $n-1$ arguments are fixed by $n-1$ elements in $R$. In this short note, starting from Brešar Theorems, we prove that a skew $n$-derivation ($n\geq 3$) on a semiprime ring $R$ must map into the center of $R$.
Let R[G] be the group ring of a group G over an associative ring R with unity such that all prime divisors of orders of elements of G are invertible in R. If R is finite and G is a Chernikov (torsion FC-) group, then each R-derivation of R[G] is inner. Similar results also are obtained for other classes of groups G and rings R.
In this article, we prove Conjecture $1$ posed in 2013 by Fo$\check{s}$ner \cite{fos} and Conjecture $1$ posed in 2014 by Ali and Fo$\check{s}$ner \cite{ali} related to generalized $(m,n)$-Jordan centralizer and derivation respectively.
In this paper, we show that if the product $(D_1D_2, d_1d_2)$ of generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on an algebra $A$ is a generalized derivation, then $d_1D_2$ and $d_2D_1$ map $A$ into $\hbox{rad}(A)$. Also, for generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on a prime ring with characteristic different from two, we give necessary and sufficient conditions under which $(D_1^2+D_1D_2,d_1^2+d_1d_2)$ is a generalized derivation as well.
In this paper, firstly as a short note, we prove that a left derivation of a semiprime $Γ$-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 $Γ$-ring with a strong commutativity preserving derivation on itself must be commutative and that a strong commutativity preserving endomorphism on a semiprime $Γ$-ring $M$ must have the form $σ(x)=x+ζ(x)$ where $ζ$ is a map from $M$ into its center, which extends some results by Bell and Daif to semiprime $Γ$-rings.
In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fošner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fošner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivati…
Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]…
Let R be a 2 torsion free semiprime ring and d a nonzero derivation. Further let A = O(R) be the orthogonal completion of R and B = B(C) the Boolean ring of C where C be the extended centroid of R. We show that if a[[d(x),x]^n- [y, d(y)]^m]^t = 0 such that a in R for all x, y in R, where m, n, t > 0 are fixed integers, then there exists an idempotent e in B such that eA is a commutative ring and d induce a zero derivation on (1-e)A.
Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $δ:R\to R$ such that \[ [δ(x),x]\in Z(R)~~(resp.~~[δ(x),x^{\ast}]\in Z(R))\text{~for~all~}x\in R, \] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple al…
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.
Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring RP with respect to a prime ideal P of R and we prove a criterion about the D- simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author. Second we construct …
Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fixed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to its center. Moreover, in semiprime case let A = O(R) be the orthogonal completion of R and B = B(C) be the Boolian ring of C, where C is the extended centroid of R, then there exists an idempotent e in B such that eA is commutative ring and d induce a zero derivation on (1-e)A.
The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that both $\partial_x \in \mathfrak{D}$ and the ring is $\mathfrak{D}$-simple, then there is $d \in \mathfrak{D}$ such that $k[x,y]$ is $\{\partial_x,d\}$-simple. As applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.
The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.
In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.
