Related papers: Jordan derivations on triangular matrix rings

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…

Let ${\mathcal{T}}$ be a triangular algebra. We say that $D=\{D_{n}: n\in N\}\subseteq L({\mathcal{T}})$ is a Jordan higher derivable mapping at $G$ if $D_{n}(ST+TS)=\sum_{i+j=n}(D_{i}(S)D_{j}(T)+D_{i}(T)D_{j}(S))$ for any $S,T\in…

Let $R$ be any ring containing a non-tivial idempotent element $e$. Let $\Im: R\rightarrow R$ be a mapping such that $\Im(ab)=\Im(b)a+b\Im(a)$ for all $a,b\in R$. In this note, our aim is to show that under some suitable restrictions…

Let $R$ be a ring with identity, $M,N$ right modules over $R$. An additive mapping $\delta$ from $R$ to $R$ is called derivation on ring $R$ if it satisfies the Leibniz condition. If $\delta$ is a derivation on $R$ and $f:M \rightarrow N$…

In this short note we prove that every Jordan derivation of triangular algebras is a derivation.

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 $\Delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More…

In this paper, we address the additivity of $n$-multiplicative isomorphisms and $n$-multiplicative derivations on Gamma rings. We proved that, if $\M$ is a $\Gamma$-ring satisfying the some conditions, then any $n$-multiplicative…

Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan…

The additivity of traces in certain tensor triangulated categories for endomorphisms of finite order of distinguished triangles is investigated. For the identity endomorphism this has been fully established by J. P. May ("The additivity of…

Let $\mathcal{U}=\left[ \begin{array}{cc} \mathcal{A} & \mathcal{M} \mathcal{N}& \mathcal{B} \end{array} \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$ are 2-torsion free. We prove that if $\phi…

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,…

Let $R$ be a ring with involution containing a nontrivial symmetric idempotent element $e$. Let $\delta: R\rightarrow R$ be a mapping such that $\delta(ab)=\delta(b)a^{\ast}+b^{\ast}\delta(a)$ for all $a,b\in R$, we call $\delta$ a…

In this note, we prove that any Jordan derivation on the generalized matrix ring $T_n(R,M)$ is a derivation. This extends some well-known results of this branch due to Bre\v{s}ar et al. in the cited literature.

Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not necessarily with the identity elements). A bijective map $\varphi$ of $\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative isomorphism} if {$\varphi…

We introduce the notion of a Jordan triple module and determine the precise conditions under which every derivation from a JB*-triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or…

For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of…

In the present paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major…

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and…

A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on…

A map $\phi$ on an associative ring is called a multiplicative Lie derivation if $\phi([x,y])=[\phi(x),y]+[x,\phi(y)]$ holds for any elements $x,y$, where $[x,y]=xy-yx$ is the Lie product. In the paper, we discuss the multiplicative Lie…