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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]

Let R be a prime ring of characteristic different from 2, Q r its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, a ∈ R a non-zero element of R, m , r ≥ 1 and s , t ≥ 0 fixed integers. If a (F (u) r (F (u) s u + u F (u) s) F (u) t) m = 0 for all u ∈ L , then either there exists b ∈ Q r such that F (x) = b x , for any x ∈ R , with ab = 0 or R ⊆ M 2 (K) , the ring of 2 × 2 matrices over a field K. [ABSTRACT FROM AUTHOR]

Let R be a unitary prime ring with the maximal left ring of quotients Q m l (R). We completely characterize the forms of additive maps F 1 , F 2 , G 1 , G 2 : R → Q m l (R) such that F 1 (x) y + F 2 (y) x + x G 1 (y) + y G 2 (x) = 0 whenever x , y ∈ R satisfy xy = 0 = yx under the assumptions that R contains a nontrivial idempotent and characteristic of R is not 2. This partially generalizes a result of T.K. Lee in [Bi-additive maps of ζ-Lie product type vanishing on zero products of XY and Y X. Comm Algebra. 2017;45(8):3449–3467]. [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]

Let R be a prime algebra over a commutative ring K, let I be a nonzero ideal of R and let f (X 1 , ... , X k) be a multilinear polynomial over K in k non-commuting indeterminates which is not central-valued on R. Suppose that a ∈ R and d is a derivation of R such that a (d (f (x 1 , ... , x k)) s − γ f (x 1 , ... , x k ) t ) m ∈ Z (R) for all x 1 , ... , x k ∈ I , where γ ∈ K , s, t, m are fixed positive integers and Z (R) is the centre of R. It is shown that a = 0 or d = 0 except when R ⊆ M 2 (F) , the 2 × 2 matrix ring over a field F. This result gives a natural generalization of several well-known theorems in the literature. Moreover, with this we give an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator conditions on Lie ideals in prime rings. J Algebra Appl. 2020;19:2050025]. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R and f (x 1 , ... , x n) a multilinear polynomial over C. Suppose that f (x 1 , ... , x n) is not central valued on R, and R does not embed in M 2 (K) , the algebra of 2 × 2 matrices over a field K. We describe all possible forms of F and G in the case [ F (G (f (r 1 , ... , r n))) , f (r 1 , ... , r n) ] = 0 for any r 1 , ... , r n ∈ R. [ABSTRACT FROM AUTHOR]

Let T be a triangular n-matrix ring ( n ≥ 2). It is shown that, under some mild assumptions, a map δ : T → T is a multiplicative Lie derivation if and only if δ (X) = d (X) + γ (X) holds for all X ∈ T , where d : T → T is an additive derivation and γ : T → Z (T) is a central-valued map vanishing on all commutators. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a description of the derivatives in U T M n (S) , the semiring of upper triangular matrices over an additively idempotent semiring S. The main result states that an arbitrary derivation in the semiring U T M n (S) is a linear combination of finite number of well-known derivations. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations on the so-called infinite dimensional generalized Witt algebras. Firstly, we prove that every 2-local derivation on the generalized Witt algebra W n (F) over the vector space F n is a derivation, where F is a field of characteristic zero. Further we consider generalized Witt algebras of the form W = W (G , I) over the field F , where I is an infinite index set and G = ⨁ i ∈ I Z = (a i) i ∈ I : a i = 0 e x c e p t f o r a f i n i t e n u m b e r o f i ∈ I , and prove that all 2-local derivations on W (G , I) are also derivations. Finally, we show that every 2-local derivation on B (Z n , I) , the Borel subalgebra of W n (F) , is a derivation. [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a finite preordered set, R a 2-torsion free commutative ring with unity and I (X , R) the incidence algebra of X over R. In this paper we prove that every nonlinear Lie derivation of I (X , R) is of the standard form. More explicitly, each nonlinear Lie derivation of I (X , R) is a sum of an inner derivation, a transitive induced derivation, an additive induced derivation and a central-valued map. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with extended centroid C and with maximal left ring of quotients Q m l (R). An additive map δ : R → Q m l (R) is called a weak Jordan derivation if δ (x 2) − δ (x) x − x δ (x) ∈ C for all x ∈ R. Applying the theory of functional identities and dealing with the low dimensional cases, we give a complete characterization of weak Jordan derivations of prime rings. Moreover, we generalize Brešar's theorem concerning additive maps δ : R → R C + C satisfying [ δ (x 2) − x δ (x) − δ (x) x , x ] = 0 for all x ∈ R. [ABSTRACT FROM AUTHOR]

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation. [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]

Let R be a prime ring of characteristic different from 2, C its extended centroid and L a non-central Lie ideal of R. If F : R → R and G : R → R are two non-zero generalized derivations of R such that [ F (u) , G (u) ] = 0 for all u ∈ L , then G = λ F for some λ ∈ C except possibly when R satisfies the standard identity s 4 of degree 4. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric 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. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation. [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]

Letbe a generalized matrix ring, whereandare 2-torsion free. We prove that ifis an additive mapping such thatwheneverthen, whereis a Jordan derivation andis a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann algebras. [ABSTRACT FROM PUBLISHER]

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields. [ABSTRACT FROM PUBLISHER]

Letbe a triangular algebra and letbe a nonlinear generalized Lie derivation with an associated nonlinear mapf. It is showedandfare the sum of an additive generalized derivation and a central map vanishing on each commutator, respectively. As applications, nonlinear generalized Lie derivations on some classical triangular algebras are characterized. [ABSTRACT FROM PUBLISHER]

Letbe a field andQa finite simply laced quiver without oriented cycles. Firstly, we prove that each Lie derivation of the generalized one-point extension of path algebrais of the standard form, and the standard decomposition is unique. Secondly, we calculated the-dimensions of all the Hochschild cohomology of the generalized one-point extension oftype path algebra, and gave a description of the cup product in the Hochschild cohomology ring. [ABSTRACT FROM PUBLISHER]

LetRbe a noncommutative prime ring with the extended centroidC,Ia nonzero ideal ofRandgab-generalized derivation ofR. We show that, iffor all, wherem, n, kare fixed positive integers, then there existssuch thatfor allunless, thematrix ring over the Galois fieldGF(2) of two elements. This gives a natural generalization of the results for derivations, generalized derivations and generalized-derivations with an X-inner automorphism. [ABSTRACT FROM PUBLISHER]

LetAbe a unital algebra over a fieldwith char, and letbe linear maps. We say thatfis a-derivation iffor all, and we say thatfis a Jordan-derivation iffor all(here,). We show that if the property that every Jordan-derivation is a-derivation holds inA, then so does in the algebrafor every commutative unital algebraS. We also show that every semiprime algebraAhas this property. Combining these two results, it follows, in particular, that the classical Jordan derivations are derivations on the tensor product between a semiprime and a commutative algebra. [ABSTRACT FROM PUBLISHER]

In this paper, we prove that every biderivation of certain triangular rings is the sum of an extremal biderivation and an inner biderivation, using the notion of maximal left ring of quotients. As a consequence, we show that every biderivation of the ring of allupper triangular matrices over a unital ring () is the sum of an extremal biderivation and an inner biderivation, which extends two results of Benkovič and Ghosseiri, respectively. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to investigate Lie superderivations of superalgebras, using the theory of functional identities in superalgebras. As a consequence, we obtain a definitive result on Lie superderivations of prime superalgebras. [ABSTRACT FROM AUTHOR]

Letbe a ring with an involutionand containing a non-trivial symmetric idempotent. Under some mild conditions on, several different characterizations of multiplicative Jordan *-derivations onare given. Based on these results, the concrete form of multiplicative Jordan *-derivations on prime *-rings and factor von Neumann algebras is respectively obtained, which generalizes some known results. [ABSTRACT FROM PUBLISHER]

The aim of the paper is to give a condition on which every centralizing trace of a bilinear map on triangular algebras is commuting. As an application, we give a short proof of a result due to Xiao et al. As another application, we give a description of Lie triple isomorphisms of certain triangular algebras, which improves a result due to Xiao et al. [ABSTRACT FROM PUBLISHER]

Letbe a non-commutative prime ring withits maximal right ring of quotients. It is proved that the functional identityfor bi-additive mapsonly has the trivial solution. Basing on the result, we characterize Jordan-derivations and generalized Jordan semiderivations in terms of derivations,-derivations and semiderivations. [ABSTRACT FROM AUTHOR]

We consider the problem of describing the form Jordan-derivations of a triangular algebra. The main result states that every Jordan-derivationofis of the form, whereis a-derivation ofandis a special mapping of. We search for sufficient conditions on a triangular algebra, such that. In particular, any Jordan-derivation of a nest algebrais a-derivation and any Jordan-derivation of an upper triangular matrix algebra, whereis a commutative unital algebra, is a-derivation. [ABSTRACT FROM AUTHOR]

Letbe a semiprime ring with centre,its right Martindale quotient ring,its extended centroid,a generalized skew derivation of, associated with a non-zero skew derivationof, andfixed integers such that, for all. Thencontains a non-zero central ideal. The case of centralizers (generalized skew derivations associated with a zero skew derivation) is also studied. [ABSTRACT FROM AUTHOR]

Letbe a unital algebra with a nontrivial idempotentover a unital commutative ring. We show that under suitable assumptions, every Lie triple derivationonis of the form, whereis a derivation of,is a singular Jordan derivation ofandis a linear mapping fromto its centrethat vanishes on. As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras. [ABSTRACT FROM AUTHOR]

Letbe a unital ring with a nontrivial idempotent, and letbe an-bimodule. We say that an additive mapis derivable atiffor anywith. In this paper, we give a necessary and sufficient condition for an additive mapto be derivable atwith. Moreover, we show that ifis a prime Banach algebra with the unit, then an additive mapis derivable atwithif and only if there is a derivationsuch thatfor all. As an application, we get a full characterization of derivable maps on some reflexive algebras and von Neumann algebras with no abelian summands. In particular, we show that an additive mapis derivable at any nonzero finite rank operator if and only if it is a derivation. New equivalent characterization of derivations on these algebras are obtained. [ABSTRACT FROM AUTHOR]

Applying the extended Jacobson density theorem for rings with automorphisms and skew derivations, we characterize a generalized skew derivationof rings or Banach algebras so that the values ofon a left ideal are nilpotent. The analogous results for-algebras and standard operator algebras on Banach spaces are also described. [ABSTRACT FROM PUBLISHER]