In this paper, we show that a map $δ$ over a triangular ring $\mathcal{T}$ satisfying $δ(ab+ba)=δ(a)b+a τ(b)+δ(b)a+bτ(a)$, for all $a,b\in \mathcal{T}$ and for some maps $τ$ over $\mathcal{T}$ satisfying $τ(ab+ba)=τ(a)b+a τ(b)+τ(b)a+bτ(a)$, is additive. Also, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(ab)=T(a)b=aT(b)$, for all $a,b\in \mathcal{T}$, is additive. Further, we establish that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(ab)=2mD(a)b+2naD(b)$, for all $a,b\in \mathcal{T}$ and integers $m,n\geq 1$, then $D$ is additive.
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.
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.
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.
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 $\mathcal A$ 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.
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…
Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on this algebra are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates an abelian subgroup of $G$. Assuming that $G$ is abelian, our technique also works to obtain the classification of $G$-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic $0$ and $…
Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b +b\cdot a)$ we have that $UT_n$ is a Jordan algebra, denoted by $UJ_n=UJ_n(K)$. In this paper, we describe the set of all $\mathbb{Z}_2$-graded polynomial identities of $UJ_2$ with any nontrivial $\mathbb{Z}_2$-grading. Moreover, we describe a linear basis for the corresponding relatively free $\mathbb{Z}_2$-graded algebra.
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).
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…
The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f-biderivations as a natural "biderivation" counterpart of the notion of "f-derivations". We first show, under some conditions, that any f-derivation is a Jordan biderivation. Then, we turn to study f-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibideriv…
We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal D$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through $U(\spl_2(\Bbbk))$. Here we introduce the Verma modules and the category $\mathfrak O$ for $\mathcal D$, which have a resemblance to the similar ones in Lie theory but induced from indecomposable modules of the 0-part of the triangular decomposition. Accordingly, there is the notion of highest weight rank (hw-rk). We classify the indecomposable modules of hw…
