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Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z 2 -grading is induced by e. This paper aims to describe Lie superderivations of U . In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z 2 -grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper was to provide a characterization of nonlinear generalized Lie n -higher derivations for a certain class of triangular algebras. It was shown that, under some mild conditions, each component G r of a nonlinear generalized Lie n -higher derivation { G r } r ∈ N of the triangular algebra U could be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ( n − 1)-th commutators on U . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and A , B be unital algebras over R . Let M be a unital (A , B) -bimodule, which is faithful as a left A -module and also as a right B -module. Let T = Tri (A , M , B) be an (n − 1) -torsion free triangular algebra. In this note, we investigate the structure of local Lie n-derivations under certain mild conditions and 2-local Lie n-derivations under weaker conditions than that of local Lie n-derivations with n ≥ 3 on T . Under their corresponding conditions, we conclude that they are both standard on T . [ABSTRACT FROM AUTHOR]

The aim of this paper was to provide a characterization of nonlinear generalized Lie $ n $-higher derivations for a certain class of triangular algebras. It was shown that, under some mild conditions, each component $ G_r $ of a nonlinear generalized Lie $ n $-higher derivation $ \{G_r\}_{r\in N} $ of the triangular algebra $ \mathcal{U} $ could be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ($ n-1 $)-th commutators on $ \mathcal{U} $.

Let $ \mathcal{A} $ be a unital algebra with nontrivial idempotents. We considered $ \mathcal{A} $ as a superalgebra according to Ghahramani and Zadeh's method. We provided a description of supercommuting maps on $ \mathcal{A} $. As a consequence, we gave a description of supercommuting maps on matrix algebras, which is different from the result on commuting maps of matrix algebras. Finally, we proved that every supercommuting map on triangular algebras is a commuting map.

We give an explicit description of a diagonal map on the Bardzell resolution for any monomial algebra, and we use this diagonal map to describe the cup product on Hochschild cohomology. Then, we prove that the cup product is zero in positive degrees for triangular monomial algebras. Our proof uses the graded-commutativity of the cup product on Hochschild cohomology and does not rely on explicit computation of the Hochschild cohomology modules. [ABSTRACT FROM AUTHOR]

Let 풰 be an algebra over the field 픽. We say that 풰 is two-sided zero product determined if every bilinear functional ϕ : 풰×풰→ 픽 the following holds: if ϕ(x,y) = 0 whenever xy = yx = 0, then there exist linear functionals F1 and F2 on 풰 such that ϕ(x,y) = F1(xy)+F2(yx) for all x,y ∈풰. We show that the unital triangular algebra 풯 = 풜ℳ0 ℬ is a two-sided zero product determined algebra if and only if 풜 and ℬ are two-sided zero product determined algebras, and then we get various results about this property for generalized triangular algebras and block upper triangular matrix algebras. We also provide an application of the main result to determine the structure of commutativity preserving maps at commutative zero products on triangular algebras. We note that some of the previous results about the two-sided zero product determined property have been generalized. [ABSTRACT FROM AUTHOR]

This research examines the Jordan f-biderivations on triangular algebras by utilizing a faithful bimodule structure. Let Ω be a triangular algebra and f be an automorphism of Ω. We demonstrate that any Jordan f-biderivation ∆ of Ω may be expressed as ∆ = D + δ + τ, in which D : Ω × Ω → Ω is a f-biderivation, δ : Ω × Ω → f(q) · lΩl is a map with δ(lΩl + qΩq, lΩl + qΩq) = {0} and τ : Ω × Ω → Zf (Ω) is a fcentral map, l is a nontrivial idempotent in Ω such that qΩl = {0}, q = 1 − l. As an application, we prove that each Jordan biderivation of triangular algebras is indeed a biderivation. [ABSTRACT FROM AUTHOR]

Let = Tri(A,M, B) be a triangular algebra. In this paper, we prove that under certain conditions, every local derivation from - into itself is a derivation; every additive 2-local Lie derivation from - into itself is a Lie derivation. [ABSTRACT FROM AUTHOR]

In this paper, by using the structure properties of algebras and the technique of matrix operation, we investigate the structure of local Lie triple derivations on triangular algebras. Under some mild conditions, we prove that each local Lie triple derivation of triangular algebras can be expressed as the sum of a derivation and a linear map from the triangular algebra to its center that vanishes at Lie triple products. Finally, we apply the main result to the problem of characterizing local Lie triple derivations on an upper triangular matrix algebra. [ABSTRACT FROM AUTHOR]

Let 풰 = AMN B be a generalized matrix algebra defined by the Morita context (A,B,M,N, ΦMN, ΨNM) and 풵(풰) the center of 풰. In this paper, under some certain conditions on 풰, we prove that if F : 풰→풰 is an additive map satisfying [x,F(y)] = 0 for any x,y ∈풰 with xy = 0 = yx, then F has the form F(x) = λx + τ(x) for all x ∈풰, where λ ∈풵(풰) and τ is an additive map from 풰 into 풵(풰). Finally as its applications, we characterize Lie centralizer maps and generalized Lie derivations on 풰. Moreover we prove that the similar conclusions remain valid on full matrix algebras, triangular algebras, upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we provide a necessary and sufficient condition for a class of trivial extensions so that each Jordan derivation on such algebras can be written as the sum of a derivation and an antiderivation. Among other things, we obtain some of the known results concerning the Jordan derivations on trivial extensions with fewer assumptions. Also, as a generalization of a well-known result on triangular matrix algebras, we show that every Jordan derivation from arbitrary triangular algebra into its certain bimodule is the sum of a derivation and an antiderivation. We also apply our results for upper triangular matrix algebra 풯n(ℛ), matrix algebra ℳn(ℛ) and the nest algebra 풯 (풩) to describe the Jordan derivations into their bimodules. [ABSTRACT FROM AUTHOR]

In this study, we consider generalized Lie nn-derivations of an arbitrary triangular algebra TT through the constructed triangular algebra T0{T}_{0}, where T0{T}_{0} is constructed using the notion of maximal left (right) ring of quotients.

Let A be an unital algebra over the complex field C . A linear map ϕ from A into itself is called a Lie centralizer at a given point G ∈ A if ϕ ([ S , T ]) = [ S , ϕ (T) ] = [ ϕ (S) , T ] for all S , T ∈ A with ST = G. The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of Jordan centralizer maps from a unital algebra A into a unital A-bimodule M. We applied our results to triangular algebras. In particular, we prove that every Jordan centralizer map on a triangular algebra is a centralizer map. [ABSTRACT FROM AUTHOR]

Let be a triangular algebra over a commutative ring ℛ. In this paper, under some mild conditions on , we prove that if δ : → is a nonlinear map satisfying δ ([ [ U , V ] , W ]) = [ [ δ (U) , V ] , W ] + [ [ U , δ (V) ] , W ] + [ [ U , V ] , δ (W) ] for any U , V , W ∈ with U V W = 0. Then δ is almost additive on , that is, δ (U + V) − δ (U) − δ (V) ∈ (). Moreover, there exist an additive derivation d of and a nonlinear map τ : → () such that δ (U) = d (U) + τ (U) for U ∈ , where τ ([ [ U , V ] , W ]) = 0 for any U , V , W ∈ with U V W = 0. [ABSTRACT FROM AUTHOR]

Let T = T r i (A , M , B) be a triangular algebra where A is a unital algebra, B is an algebra which is not necessarily unital, and M is a faithful (A , B )-bimodule which is unital as a left A -module. In this paper, under some mild conditions on T , we show that if ϕ : T → T is a linear map satisfying A , B ∈ T , A B = P ⟹ ϕ ([ A , B ]) = [ A , ϕ (B) ] = [ ϕ (A) , B ] , where P is the standard idempotent of T , then ϕ = ψ + γ where ψ : T → T is a centralizer and γ : T → Z (T) is a linear map vanishing at commutators [A, B] with A B = P whrere Z (T) is the center of T . Applying our result, we characterize linear maps on T that behave like generalized Lie 2-derivations at idempotent products as an application of above result. Our results are applied to upper triangular matrix algebras and nest algebras. [ABSTRACT FROM AUTHOR]

Let A be a triangular algebra and σ be an automorphism of A . We consider the problem of describing the form of Lie σ-derivations of A . In particular, we give sufficient conditions that every Lie σ-derivation d of A is the sum d = Δ + γ , where Δ is a σ-derivation of A and γ is a linear mapping from A to its σ-centre that vanishes on A , A . As an application, Lie σ-derivations of (block) upper triangular matrix algebras and nest algebras are determined. [ABSTRACT FROM AUTHOR]

This article is about the study of n-Lie derivations on triangular algebras under certain restrictions and it is shown that every n-Lie derivation apparently split into an extremal nderivation and an n-linear central map on triangular algebras. In addition, we implement this result to the classical examples of triangular algebras. [ABSTRACT FROM AUTHOR]

Let N be the set of nonnegative integers and A be a (n - 1) -torsion free triangular algebra over a commutative ring R . In the present paper, under some mild assumptions on A , it is prove that if δ : A → A is an R -linear mapping satisfying δ (p n (X 1 , X 2 , ⋯ , X n)) = ∑ i = 1 i = n p n (X 1 , X 2 , ⋯ , X i - 1 , δ (X i) , X i + 1 , ⋯ , X n) for all X 1 , X 2 , ⋯ , X n ∈ A with X 1 X 2 = 0 (resp. X 1 X 2 = P , where P is a nontrivial idempotent of A ), then δ = d + τ ; where d : A → A is a derivation and τ : A → Z (A) (where Z (A) is the center of A ) is an R -linear map vanishing at every (n - 1) -th commutator p n (X 1 , X 2 , ⋯ , X n) with X 1 X 2 = 0 (resp. X 1 X 2 = P ). [ABSTRACT FROM AUTHOR]

Let U be a triangular algebra. Under certain assumptions, we show that every multiplicative Lie s-derivation L: U U is of the form d + t, where d is an additive s-derivation on U and t is a map from U to its center Zs(U) that vanishes on Lie product. As an application, we study the multiplicative Lie s-derivation on nest algebras and block upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on τ(N)\tau ({\mathscr{N}}) is a {g,h}-derivation if and only if dim0+≠1\dim {0}_{+}\ne 1 or dimH−⊥≠1\dim {H}_{-}^{\perp }\ne 1, where N{\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and τ(N)\tau ({\mathscr{N}}) is the associated nest algebra.

The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

In this paper, we give a description of Lie (Jordan) triple derivations and generalized Lie (Jordan) triple derivations of an arbitrary triangular algebra A through a triangular algebra A 0 , where A 0 is a triangular algebra constructed from the given triangular algebra A using the notion of maximal left (right) ring of quotients such that A is the subalgebra of A 0 having the same unity. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let R be a commutative ring with unity, A = Tri(A,M,B) be a triangular algebra consisting of unital algebras A,B and (A,B)-bimodule M which is faithful as a left A-module and also as a right B-module. In this article,we study Jordan triple (σ,τ)-higher derivation onAand prove that every Jordan triple (σ,τ)-higher derivation on A is a (σ,τ)-higher derivation on A.

Let R be a commutative ring with identity element, A and B be unital algebras over R and let M be (A,B)-bimodule which is faithful as a left A-module and also faithful as a right B-module. Suppose that A = Tri(A,M,B) is a triangular algebra which is 2-torsion free and σ, Γ be automorphisms of A. A map δ:A→A (not necessarily linear) is called a multiplicative generalized (σ, Γ)-derivation (resp. multiplicative generalized Jordan (σ, Γ)-derivation) on A associated with a (σ, Γ)-derivation (resp. Jordan (σ, Γ)-derivation) d on A if δ(xy) = δ(x)r(y) + σ(x)d(y) (resp. σ(x2) = δ(x)r(x) + δ(x)d(x)) holds for all x, y Є A. In the present paper it is shown that if δ:A→A is a multiplicative generalized Jordan (σ, Γ)-derivation on A, then δ is an additive generalized (σ, Γ)-derivation on A.

Let N be the set of nonnegative integers and A be a 2-torsion free triangular algebra over a commutative ring R. In the present paper, under some lenient assumptions on A, it is proved that if Δ = {δg}n∈N is a sequence of R-linear mappings δn: ... satisfying δn([[x, y], z]) = P i+j+k=n [[i(x), j(y)], k(z)] for all x; y; z 2 A with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of A), then for each n 2 N, δn = dn+n; where dn: A ! A is R-linear mapping satisfying dn(xy) = P i+j=n di(x)dj(y) for all x; y 2 A, i.e. D = fdngn2N is a higher derivation on A and Tn: A → Z(A) (where Z(A) is the center of A) is an R-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp. xy = p). [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of mapping problems on triangular algebras without assuming unity. We give the definitions of strong faithful bimodules, extreme centres, and extreme Lie derivations. Our main result is to give a description of Lie derivations on triangular algebras without assuming unity. As an application we shall give a characterization of Lie derivations on upper triangular matrix algebras over a faithful algebra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by Benkovič. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras. [ABSTRACT FROM AUTHOR]

In this paper we mainly characterize Lie higher derivations on triangular algebras by local action. Let T = [ A M 0 B ] be a triangular algebra over a commutative ring R and Z (T) be the center of T . Under some mild conditions on T , we prove that if a family Δ = { δ n } n = 0 ∞ of R -linear mappings on T satisfies the condition δ n ([ X , Y ]) = ∑ i + j = n [ δ i (X) , δ j (Y) ] for any X , Y ∈ T with XY = 0 (resp. XY = P, where P is a fixed nontrivial idempotent of T ), then there exist a higher derivation D = { d n } n = 0 ∞ and an R -linear mapping τ n : T → Z (T) vanishing on commutators [X, Y] with XY = 0 (resp. XY = P) such that δ n (X) = d n (X) + τ n (X) for all X ∈ T . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we construct a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients. As an application we give a description of Lie (Jordan) derivations of arbitrary triangular algebras through the constructed triangular algebra. [ABSTRACT FROM AUTHOR]

Let A be a triangular algebra. We show that under suitable assumptions every generalized Lie n-derivation F : A → A associated with a linear map D : A → A is of the form F (x) = λ x + Δ (x) , where λ ∈ Z (A) and Δ is a Lie n-derivation of A. We solve this problem using commuting and centralizing maps. We also prove that under certain mild conditions any centralizing map on a triangular algebra is commuting. As an application, we give a description of generalized Lie n-derivations on classical examples of triangular algebras: upper triangular matrix algebras and nest algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the images of non-commutative multilinear polynomials on upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in $B({\cal H})$. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let A, B be unital R-algebras over a commutative ring R with unity and M be a nonzero (A, B)-bimodule which is faithful as a left A-module and also as a right B-module. Suppose that 𝖀 = Tri(A,M, B) is a 2-torsion free triangular algebra consisting of A, B and M and N is the set of non-negative integers. In the present paper it is shown that if Δ = {δn}n∈N is a sequence of mappings δn : 𝖀 → 𝖀 (not necessarily linear) such that δn(XY Z+ZYX) = Σi+j+k=n(δi(X)δj (Y)δk(Z)+δi(Z)δj (Y)δk(X)) for all X, Y, Z ∈𝖀, then Δ = {δn}n∈N is an additive higher derivation on 𝖀. [ABSTRACT FROM AUTHOR]

In this sequel, some conditions are provided under which a Lie centralizer maps on a trivial extension algebra can be decomposed into the sum of a centralizer and a center valued map. Our results are examined for the triangular algebras. [ABSTRACT FROM AUTHOR]

In this paper, we will describe the general form of commuting mappings of Hochschild extension algebras and characterize the properness of commuting mappings on a special class of Hochschild extension algebras with the so-called p. [ABSTRACT FROM AUTHOR]

Let 풯 be a (n−1)-torsion free triangular algebra over a communicative ring ℛ and 풵(풯) be the center of 풯. Let GL:풯→풯 be a nonlinear generalized Lie n-derivation with associated Lie n-derivation L. Suppose that is the (n−1)-th commutator defined by n indeterminates . In this paper, we prove that under certain assumptions, the nonlinear generalized Lie n-derivation GL is of the form GL = δ+τ, where δ:풯→풯 is an additive generalized derivation on 풯 and τ:풯→풵(풯 ) is a mapping vanishing on each (n−1)-th commutator . Some relevant applications of this result are presented at the end of this article. [ABSTRACT FROM AUTHOR]

For a triangular algebra 풜 and an automorphism σ of 풜, we describe linear maps F,G:풜→풜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈풜 are such that xy = 0. In particular, when 풜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈풜, where D:풜→풜 is a skew derivation. When 풜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. A triangular algebra is an algebra of the form A=AM0B where A and B are unital algebras over R and M is an (A,B)-bimodule which is faithful as a left A-module as well as a right B-module. In this paper, we study nonlinear generalized Lie triple higher derivation on A and show that under certain assumptions on A, every nonlinear generalized Lie triple higher derivation on A is of standard form, i.e., each component of a nonlinear generalized Lie triple higher derivation on A can be expressed as the sum of an additive generalized higher derivation and a nonlinear functional vanishing on all Lie triple products on A. [ABSTRACT FROM AUTHOR]

Let T be a triangular algebra over a commutative ring R, ξ be an automorphism of T and Zξ(T) be the ξ-center of T. Suppose that q:T×T→T is an R-bilinear mapping and that Tq:T→T is a trace of q. The aim of this article is to describe the form of Tq satisfying the commuting condition [Tq(x),x]ξ=0 (resp. the centralizing condition [Tq(x),x]ξ∈Zξ(T)) for all x∈T. More precisely, we will consider the question of when Tq satisfying the previous condition has the so-called proper form. [ABSTRACT FROM AUTHOR]

The aim of the paper is to prove that every Jordan-derivation of a triangular algebra that exists either a left weak loyal bimodule or a right weak loyal bimodule, is a-derivation. As an application we show that every Jordan-derivation of a(block) upper triangular matrix algebra, where, is a-derivation, which improves some results given by Benkovič in 2016. [ABSTRACT FROM AUTHOR]